Theorem bj-ax12v3ALT 32676

Description: Alternate proof of bj-ax12v3 32675. Uses axc11r 2187 and axc15 2303 instead of ax-12 2047. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ax12v3ALT	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem bj-ax12v3ALT

Step	Hyp	Ref	Expression
1		ax-5 1839	. . . 4 |- ( ph -> A. y ph )
2		axc11r 2187	. . . 4 |- ( A. x x = y -> ( A. y ph -> A. x ph ) )
3		ala1 1741	. . . 4 |- ( A. x ph -> A. x ( x = y -> ph ) )
4	1, 2, 3	syl56 36	. . 3 |- ( A. x x = y -> ( ph -> A. x ( x = y -> ph ) ) )
5	4	a1d 25	. 2 |- ( A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
6		axc15 2303	. 2 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
7	5, 6	pm2.61i 176	1 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)