Theorem vtocl2 3261

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 3209	. . . . 5 |- E. x x = A
3		vtocl2.2	. . . . . 6 |- B e. _V
4	3	isseti 3209	. . . . 5 |- E. y y = B
5		eeanv 2182	. . . . . 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 219	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ph -> ps ) )
8	7	2eximi 1763	. . . . . 6 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( ph -> ps ) )
9	5, 8	sylbir 225	. . . . 5 |- ( ( E. x x = A /\ E. y y = B ) -> E. x E. y ( ph -> ps ) )
10	2, 4, 9	mp2an 708	. . . 4 |- E. x E. y ( ph -> ps )
11		19.36v 1904	. . . . 5 |- ( E. y ( ph -> ps ) <-> ( A. y ph -> ps ) )
12	11	exbii 1774	. . . 4 |- ( E. x E. y ( ph -> ps ) <-> E. x ( A. y ph -> ps ) )
13	10, 12	mpbi 220	. . 3 |- E. x ( A. y ph -> ps )
14	13	19.36iv 1905	. 2 |- ( A. x A. y ph -> ps )
15		vtocl2.4	. . 3 |- ph
16	15	ax-gen 1722	. 2 |- A. y ph
17	14, 16	mpg 1724	1 |- ps

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  caovord  6845  sornom  9099  wloglei  10560  ipodrsima  17165  mpfind  19536  mclsppslem  31480  monotoddzzfi  37507