Theorem vtocl2 2767

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl2.1	|- A e. _V
vtocl2.2	|- B e. _V
vtocl2.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
vtocl2.4	|- ph

Assertion

Ref	Expression
vtocl2	|- ps

Distinct variable groups:   x, y, A   x, B, y   ps, x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . . 6 |- A e. _V
2	1	isseti 2720	. . . . 5 |- E. x x = A
3		vtocl2.2	. . . . . 6 |- B e. _V
4	3	isseti 2720	. . . . 5 |- E. y y = B
5		eeanv 1912	. . . . . 6 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
6		vtocl2.3	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	6	biimpd 143	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ph -> ps ) )
8	7	2eximi 1581	. . . . . 6 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( ph -> ps ) )
9	5, 8	sylbir 134	. . . . 5 |- ( ( E. x x = A /\ E. y y = B ) -> E. x E. y ( ph -> ps ) )
10	2, 4, 9	mp2an 423	. . . 4 |- E. x E. y ( ph -> ps )
11		nfv 1508	. . . . 5 |- F/ y ps
12	11	19.36-1 1653	. . . 4 |- ( E. y ( ph -> ps ) -> ( A. y ph -> ps ) )
13	10, 12	eximii 1582	. . 3 |- E. x ( A. y ph -> ps )
14	13	19.36aiv 1881	. 2 |- ( A. x A. y ph -> ps )
15		vtocl2.4	. . 3 |- ph
16	15	ax-gen 1429	. 2 |- A. y ph
17	14, 16	mpg 1431	1 |- ps

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712

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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714

This theorem is referenced by:  undifexmid  4154  caovord  5989  ctssexmid  7087