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 -> ( A. y ph -> A. x ( x = y -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		axc11r	|- ( A. x x = y -> ( A. y ph -> A. x ph ) )
2		ala1	|- ( A. x ph -> A. x ( x = y -> ph ) )
3	1 2	syl6	|- ( A. x x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
4	3	a1d	|- ( A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
5		sp	|- ( A. y ph -> ph )
6		axc15	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
7	5 6	syl7	|- ( -. A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
8	4 7	pm2.61i	|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )