Theorem vtocl2 2815

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 2768	. . . . 5 |- E. x x = A
3		vtocl2.2	. . . . . 6 |- B e. _V
4	3	isseti 2768	. . . . 5 |- E. y y = B
5		eeanv 1948	. . . . . 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 144	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ph -> ps ) )
8	7	2eximi 1612	. . . . . 6 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( ph -> ps ) )
9	5, 8	sylbir 135	. . . . 5 |- ( ( E. x x = A /\ E. y y = B ) -> E. x E. y ( ph -> ps ) )
10	2, 4, 9	mp2an 426	. . . 4 |- E. x E. y ( ph -> ps )
11		nfv 1539	. . . . 5 |- F/ y ps
12	11	19.36-1 1684	. . . 4 |- ( E. y ( ph -> ps ) -> ( A. y ph -> ps ) )
13	10, 12	eximii 1613	. . 3 |- E. x ( A. y ph -> ps )
14	13	19.36aiv 1913	. 2 |- ( A. x A. y ph -> ps )
15		vtocl2.4	. . 3 |- ph
16	15	ax-gen 1460	. 2 |- A. y ph
17	14, 16	mpg 1462	1 |- ps

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762

This theorem is referenced by:  undifexmid  4222  caovord  6090  ctssexmid  7209