Theorem ax12vALT 2131

Description: Alternate proof of ax12v 2130 that avoids theorem axc16 2011 and is proved directly from ax-12 1792 rather than via axc15 2035. (Contributed by Jim Kingdon, 15-Dec-2017.) (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

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1710	. 2 |- E. z z = y
2		ax-5 1670	. . . . 5 |- ( ph -> A. z ph )
3		ax-12 1792	. . . . 5 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
4	2, 3	syl5 32	. . . 4 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
5		equequ2 1737	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
6	5	imbi1d 317	. . . . . . 7 |- ( z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
7	6	albidv 1679	. . . . . 6 |- ( z = y -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
8	7	imbi2d 316	. . . . 5 |- ( z = y -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
9	5, 8	imbi12d 320	. . . 4 |- ( z = y -> ( ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) <-> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
10	4, 9	mpbii 211	. . 3 |- ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
11	10	exlimiv 1688	. 2 |- ( E. z z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
12	1, 11	ax-mp 5	1 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1367   = wceq 1369   E.wex 1586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792

This theorem depends on definitions:  df-bi 185  df-ex 1587

This theorem is referenced by: (None)