Theorem ax12vALT 2153

Description: Alternative proof of ax12v 1804, shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax12vALT	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax12vALT

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 |- ( ph -> ( x = y -> ph ) )
2		axc16 1888	. . . 4 |- ( A. x x = y -> ( ( x = y -> ph ) -> A. x ( x = y -> ph ) ) )
3	1, 2	syl5 32	. . 3 |- ( A. x x = y -> ( ph -> A. x ( x = y -> ph ) ) )
4	3	a1d 25	. 2 |- ( A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
5		axc15 2058	. 2 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
6	4, 5	pm2.61i 164	1 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600

This theorem is referenced by: (None)