Metamath Proof Explorer


Theorem ax12

Description: Rederivation of Axiom ax-12 from ax12v (used only via sp ) , axc11r , and axc15 (on top of Tarski's FOL). Since this version depends on ax-13 , usage of the weaker ax12v , ax12w , ax12i are preferred. (Contributed by NM, 22-Jan-2007) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020) (Proof shortened by BJ, 4-Jul-2021) (New usage is discouraged.)

Ref Expression
Assertion ax12 x = y y φ x x = y φ

Proof

Step Hyp Ref Expression
1 axc11r x x = y y φ x φ
2 ala1 x φ x x = y φ
3 1 2 syl6 x x = y y φ x x = y φ
4 3 a1d x x = y x = y y φ x x = y φ
5 sp y φ φ
6 axc15 ¬ x x = y x = y φ x x = y φ
7 5 6 syl7 ¬ x x = y x = y y φ x x = y φ
8 4 7 pm2.61i x = y y φ x x = y φ