Corbyn then began to contrast her inaugural speech outside Number 10 last Wednesday with the Government's record, and demanded a public inquiry into the events around the so-called 'battle of Orgreave' between miners and the police in 1984.
However, May brushed off the question and said her new Home Secretary, Amber Rudd, would answer later today.
May also made hay with Labour divisions by congratulating the 140 Labour MPs who backed renewal of Trident on Monday, in defiance of their leader's position on the issue.
May answered the substance of the question regarding housing but omitted to reply to Corbyn's challenge regarding Johnson.
Corbyn then turned to the subject of austerity and criticised the effect it had had on some of the poorest in society and referenced "job insecurity" to jeers from MPs.
Awkward question from Corbyn on Boris Johnson's suitability to be foreign sec - which May ignored #PMQs— Westminster Advisers (@WA_Comms) July 20, 2016
May used the opportunity to highlight the practices of unscrupulous bosses in a thinly veiled attack on Corbyn's leadership of the Labour Party, drawing comparisons with Margaret Thatcher from some commentators.
Backbenchers poking fun at Corbyn talking about 'job insecurity', as he is having to fight for his own position— Laura Kuenssberg (@bbclaurak) July 20, 2016
Peak Thatcher https://t.co/IEZrLbXwTb— (((Wil Barber))) (@wilbarber) July 20, 2016
"An unscrupulous boss who doesn't listen to his workers and exploits the rules to further his own career" - May's assessment of Corbyn— Westminster Advisers (@WA_Comms) July 20, 2016
After May reiterated her earlier message that "Brexit means Brexit", SNP leader Angus Robertson asked May to give assurances that, given Scotland's vote in the EU referendum, "Remain would mean Remain?".
If Theresa May plays it straight, w 1 good joke & 1 unscripted moment, at #PMQs she'll be hailed as greatest parliamentarian since Disraeli— John Rentoul (@JohnRentoul) July 20, 2016
But May was in no mood to give concessions and instead talked up the importance of the union in the UK.
Angus Robertson now, in his typical assured style, asks whether "remain means remain when it comes to Scotland?" #PMQs— Bellenden (@BellendenLtd) July 20, 2016
May says she is willing to "listen to all options put forward" by Scotland but emphasises commitment to union #PMQs— Westminster Advisers (@WA_Comms) July 20, 2016
Nusrat Ghani calls on PM to condemn the use of the word honour, when discussing what have been widely been called 'honour killings' #PMQs— Bellenden (@BellendenLtd) July 20, 2016
There was also a question around the terrorist attacks in Nice last week and how the continuing problem of terrorism might be addressed now that Britain has left the EU.
May pledges crack down on terrorism at all levels and says there's "absolutely no honour" in so-called honour kilings #PMQs— Westminster Advisers (@WA_Comms) July 20, 2016
May told the House that although Britain was leaving the EU, it would not leave Europe and would continue to co-operate on issues such as terrorism.
Conservative backbencher Stuart Andrew drew laughter with his question, designed to highlight his party's record on social mobility.
Commentators gave May's first performance at the dispatch box a broadly positive review...
May makes case for Blue Collar Conservatism by drawing on backgrounds of several of her Tory MPs #PMQs— Westminster Advisers (@WA_Comms) July 20, 2016
...in contrast to Corbyn's performance.
Immediate reaction seems to be May gave a competent and assure first performance at #PMQs— Bellenden (@BellendenLtd) July 20, 2016
Corbyn essentially asking angry versions of questions usually planted by Govt whips. Too vague and too easy for May to counter.— (((Wil Barber))) (@wilbarber) July 20, 2016
According to Brandwatch, the final PMQs before the summer recess drew 19,000 tweets, with peaks for May's barbs about unscrupulous bosses and Trident.
No humour. No wit. Can't think on his feet. Can't do sustained questions. Utterly, utterly useless as leader of the opposition.— Theo Bertram (@theobertram) July 20, 2016
Tweets about May were 48 per cent positive and 52 per cent negative while for Corbyn they were 39 per cent positive and 61 per cent negative.