Theorem ax12v2OLD 2342

Description: Obsolete proof of ax12v 2048 as of 24-Mar-2021. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ax12v2OLD.1	|- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )

Assertion

Ref	Expression
ax12v2OLD	|- ( -. 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 ax12v2OLD

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 |- E. z z = y
2		dveeq2 2298	. . . 4 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
3		ax12v2OLD.1	. . . . 5 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
4		equequ2 1953	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
5	4	sps 2055	. . . . . 6 |- ( A. x z = y -> ( x = z <-> x = y ) )
6		nfa1 2028	. . . . . . . 8 |- F/ x A. x z = y
7	5	imbi1d 331	. . . . . . . 8 |- ( A. x z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
8	6, 7	albid 2090	. . . . . . 7 |- ( A. x z = y -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
9	8	imbi2d 330	. . . . . 6 |- ( A. x z = y -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
10	5, 9	imbi12d 334	. . . . 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 223	. . . 4 |- ( A. x z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
12	2, 11	syl6 35	. . 3 |- ( -. A. x x = y -> ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
13	12	exlimdv 1861	. 2 |- ( -. A. x x = y -> ( E. z z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
14	1, 13	mpi 20	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

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:  ax12a2OLD  2343