Isn't the inference that mathematics is inconsistent?
A proposition is a statement which is either true or false (and not both, and not neither).
An axiom is a proposition which we assume to be true, "for free".
A logical deduction is a way to combine or modify true statements to get other true statements.
A proof of a statement is a finite sequence of propositions. The sequence starts with an axiom, and ends with the target statement. Its other elements are propositions---these can be axioms or not. Each proposition which (i) is in the sequence and (ii) is not an axiom, must be the result of applying a logical deduction to one or more propositions which appear before it in the sequence.
A theorem is a proposition for which there exists a proof.
If we can prove two theorems which contradict one another (like Φ and ¬Φ in the quote above), then we say that our system for finding proofs is inconsistent. Since the OP starts out assuming that mathematics is inconsistent and ends up producing two contradictory theorems, he does not seem to get anywhere far. And his inference---that mathematics is consistent---seems to be incorrect.
What am I missing here?
From that assumption, they derived a contradiction, therefore the assumption can't be true, and math must be consistent.
This is what I don't see. To have derived a contradiction, they should have shown: "mathematics is consistent". Instead they ended up showing (because Φ and ¬Φ): "mathematics is inconsistent". Which is what they started out with. Where is the contradiction?
Hmm.. while writing this out, I think I am starting to get the source of the confusion. "Mathematics is inconsistent" is a meta statement about mathematics, and "Φ and ¬Φ" is a "plain" statement within the said mathematics. My intuitive feeling is that the latter contradiction does not "affect" the meta statement. I need to think further to make this more rigorous.
OT: Why the downvote(s)? Does my comment take away from the discussion?
I elucidate it in another comment: