Metamath Proof Explorer


Theorem ax12fromc15

Description: Rederivation of Axiom ax-12 from ax-c15 , ax-c11 (used through dral1-o ), and other older axioms. See Theorem axc15 for the derivation of ax-c15 from ax-12 .

An open problem is whether we can prove this using ax-c11n instead of ax-c11 .

This proof uses newer axioms ax-4 and ax-6 , but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 and ax-c10 . (Contributed by NM, 22-Jan-2007) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion ax12fromc15 x = y y φ x x = y φ

Proof

Step Hyp Ref Expression
1 biidd x x = y φ φ
2 1 dral1-o x x = y x φ y φ
3 ax-1 φ x = y φ
4 3 alimi x φ x x = y φ
5 2 4 syl6bir x x = y y φ x x = y φ
6 5 a1d x x = y x = y y φ x x = y φ
7 ax-c5 y φ φ
8 ax-c15 ¬ x x = y x = y φ x x = y φ
9 7 8 syl7 ¬ x x = y x = y y φ x x = y φ
10 6 9 pm2.61i x = y y φ x x = y φ