Theorem ax12v2-o 34234

Description: Rederivation of ax-c15 34174 from ax12v 2048 (without using ax-c15 34174 or the full ax-12 2047). Thus, the hypothesis (ax12v 2048) provides an alternate axiom that can be used in place of ax-c15 34174. See also axc15 2303. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
ax12v2-o	|- ( -. 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-o

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 |- E. z z = y
2		ax12v2-o.1	. . . . 5 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
3		equequ2 1953	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
4	3	adantl 482	. . . . . 6 |- ( ( -. A. x x = y /\ z = y ) -> ( x = z <-> x = y ) )
5		dveeq2-o 34218	. . . . . . . . 9 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
6	5	imp 445	. . . . . . . 8 |- ( ( -. A. x x = y /\ z = y ) -> A. x z = y )
7		nfa1-o 34200	. . . . . . . . 9 |- F/ x A. x z = y
8	3	imbi1d 331	. . . . . . . . . 10 |- ( z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
9	8	sps-o 34193	. . . . . . . . 9 |- ( A. x z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
10	7, 9	albid 2090	. . . . . . . 8 |- ( A. x z = y -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
11	6, 10	syl 17	. . . . . . 7 |- ( ( -. A. x x = y /\ z = y ) -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
12	11	imbi2d 330	. . . . . 6 |- ( ( -. A. x x = y /\ z = y ) -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
13	4, 12	imbi12d 334	. . . . 5 |- ( ( -. A. x x = y /\ z = y ) -> ( ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) <-> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
14	2, 13	mpbii 223	. . . 4 |- ( ( -. A. x x = y /\ z = y ) -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
15	14	ex 450	. . 3 |- ( -. A. x x = y -> ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
16	15	exlimdv 1861	. 2 |- ( -. A. x x = y -> ( E. z z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
17	1, 16	mpi 20	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   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-11 2034  ax-12 2047  ax-13 2246  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  ax12a2-o  34235