Truth table for p -> q
p q | p -> q
------
T T | T
T F | F
F T | T
F F | T
Truth table for q -> p
p q | q -> p
------
T T | T
T F | T
F T | F
F F | T
Of course unpacking "women menstruate" involves predicate logic too. Which is beyond the scope of my keyboard skills, but roughly speaking
The universal
All women have XX chromosomes
implies
If Kate is a woman then Kate has XX chromosomes
whereas the only statement
Only women menstruate
is more complicatedly a conditional nested inside the quantifier, and implies
For all individuals (if this individual menstruates, then this individual is a woman)
(note that it's flipped the direction of the if-then arrow: hence my comment about affirming the consequent). Anyhow this in turn implies (instantiation)
If Kate menstruates, Kate is a woman.
The actual proof would take a bit of finessing and a few more lines.
But many of the TRAs supposed "counter-examples" are in fact instances of them "affirming the consequent". And equally importantly, they just don't get the difference between soundness (the rules don't generate internal inconsistencies) and truth of its premises.
Many of them can't get their heads round the fact that their elaborately constructed arguments fall at the first hurdle if one does not accept TWAW.
(Conversely, the Labour leadership candidate who said "well of course this convicted TW sex offender should go in a women's prison, because she is a woman" may look like an example of a reductio ad absurdum of her TWAW premise to us, but looks both sound and truth-generating to the politician. If you don't buy that the end state truly is self-evidently barking mad, then it can't be used as a reductio).