Theorem ax11v2 1748

Description: Recovery of ax11o 1750 from ax11v 1755 without using ax-11 1442. The hypothesis is even weaker than ax11v 1755, 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 1750. (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 1631	. 2 |- E. z z = y
2		ax11v2.1	. . . . 5 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
3		equequ2 1646	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
4	3	adantl 271	. . . . . 6 |- ( ( -. A. x x = y /\ z = y ) -> ( x = z <-> x = y ) )
5		dveeq2 1743	. . . . . . . . 9 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
6	5	imp 122	. . . . . . . 8 |- ( ( -. A. x x = y /\ z = y ) -> A. x z = y )
7		hba1 1478	. . . . . . . . 9 |- ( A. x z = y -> A. x A. x z = y )
8	3	imbi1d 229	. . . . . . . . . 10 |- ( z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
9	8	sps 1475	. . . . . . . . 9 |- ( A. x z = y -> ( ( x = z -> ph ) <-> ( x = y -> ph ) ) )
10	7, 9	albidh 1414	. . . . . . . 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 228	. . . . . 6 |- ( ( -. A. x x = y /\ z = y ) -> ( ( ph -> A. x ( x = z -> ph ) ) <-> ( ph -> A. x ( x = y -> ph ) ) ) )
13	4, 12	imbi12d 232	. . . . 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 146	. . . 4 |- ( ( -. A. x x = y /\ z = y ) -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
15	14	ex 113	. . 3 |- ( -. A. x x = y -> ( z = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) )
16	15	exlimdv 1747	. 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 102   <-> wb 103   A.wal 1287   E.wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  ax11a2  1749