Theorem vtocl2 2340

Description: Implicit substitution of classes for set variables. (The proof was 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

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . . 6 |- A e. _V
2	1	isseti 2297	. . . . 5 |- E.x x = A
3		vtocl2.2	. . . . . 6 |- B e. _V
4	3	isseti 2297	. . . . 5 |- E.y y = B
5		eeanv 1707	. . . . . 6 |- (E.xE.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 170	. . . . . . 7 |- ((x = A /\ y = B) -> (ph -> ps))
8	7	2eximi 1388	. . . . . 6 |- (E.xE.y(x = A /\ y = B) -> E.xE.y(ph -> ps))
9	5, 8	sylbir 218	. . . . 5 |- ((E.x x = A /\ E.y y = B) -> E.xE.y(ph -> ps))
10	2, 4, 9	mp2an 761	. . . 4 |- E.xE.y(ph -> ps)
11		19.36v 1679	. . . . 5 |- (E.y(ph -> ps) <-> (A.yph -> ps))
12	11	exbii 1398	. . . 4 |- (E.xE.y(ph -> ps) <-> E.x(A.yph -> ps))
13	10, 12	mpbi 206	. . 3 |- E.x(A.yph -> ps)
14	13	19.36aiv 1680	. 2 |- (A.xA.yph -> ps)
15		vtocl2.4	. . 3 |- ph
16	15	ax-gen 1305	. 2 |- A.yph
17	14, 16	mpg 1332	1 |- ps

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  caoprcom 4986  caoprord 4989  ersym 5330  ersym2 16256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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