Theorem ax11v2 1808

Description: Recovery of ax11o 1810 from ax11v 1815 without using ax-11 1494. The hypothesis is even weaker than ax11v 1815, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1810. (Contributed by NM, 2-Feb-2007.)

Hypothesis

Ref	Expression
ax11v2.1	|- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )

Assertion

Ref	Expression
ax11v2	|- ( -. 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 ax11v2

Step	Hyp	Ref	Expression
1		a9e 1684	. 2 |- E. z z = y
2		ax11v2.1	. . . . 5 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
3		equequ2 1701	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
4	3	adantl 275	. . . . . 6 |- ( ( -. A. x x = y /\ z = y ) -> ( x = z <-> x = y ) )
5		dveeq2 1803	. . . . . . . . 9 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
6	5	imp 123	. . . . . . . 8 |- ( ( -. A. x x = y /\ z = y ) -> A. x z = y )
7		hba1 1528	. . . . . . . . 9 |- ( A. x z = y -> A. x A. x z = y )
8	3	imbi1d 230	. . . . . . . . . 10 |- ( z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
9	8	sps 1525	. . . . . . . . 9 |- ( A. x z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
10	7, 9	albidh 1468	. . . . . . . 8 |- ( A. x z = y -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
11	6, 10	syl 14	. . . . . . 7 |- ( ( -. A. x x = y /\ z = y ) -> ( A. x ( x = z -> ph ) <-> A. x ( x = y -> ph ) ) )
12	11	imbi2d 229	. . . . . 6 |- ( ( -. A. x x = y /\ z = y ) -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
13	4, 12	imbi12d 233	. . . . 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 147	. . . 4 |- ( ( -. A. x x = y /\ z = y ) -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
15	14	ex 114	. . 3 |- ( -. A. x x = y -> ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
16	15	exlimdv 1807	. 2 |- ( -. A. x x = y -> ( E. z z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
17	1, 16	mpi 15	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  ax11a2  1809