Theorem ax12v2 2084

Description: Recovery of ax-c15 2221 from ax12v 1856. This proof uses ax-c11n 2220 and ax-12 1855. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

Hypothesis

Ref	Expression
ax12v2.1	|- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )

Assertion

Ref	Expression
ax12v2	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Distinct variable groups:   x, z   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax12v2

Step	Hyp	Ref	Expression
1		ax6ev 1750	. 2 |- E. z z = y
2		dveeq2 2043	. . . 4 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
3		ax12v2.1	. . . . 5 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
4		equequ2 1800	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
5	4	sps 1866	. . . . . 6 |- ( A. x z = y -> ( x = z <-> x = y ) )
6		nfa1 1898	. . . . . . . 8 |- F/ x A. x z = y
7	5	imbi1d 317	. . . . . . . 8 |- ( A. x z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
8	6, 7	albid 1886	. . . . . . 7 |- ( A. x z = y -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
9	8	imbi2d 316	. . . . . 6 |- ( A. x z = y -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
10	5, 9	imbi12d 320	. . . . 5 |- ( A. x z = y -> ( ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) <-> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
11	3, 10	mpbii 211	. . . 4 |- ( A. x z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
12	2, 11	syl6 33	. . 3 |- ( -. A. x x = y -> ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
13	12	exlimdv 1725	. 2 |- ( -. A. x x = y -> ( E. z z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
14	1, 13	mpi 17	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   A.wal 1393   E.wex 1613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618

This theorem is referenced by:  ax12a2  2085