Theorem vtocl2 1839

Description: Implicit substitution of classes for set variables.

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	. . . . 5 |- A e. V
2	1	isseti 1811	. . . 4 |- E.x x = A
3		vtocl2.2	. . . . 5 |- B e. V
4	3	isseti 1811	. . . 4 |- E.y y = B
5		eeanv 1321	. . . . 5 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
6		vtocl2.3	. . . . . . 7 |- ((x = A /\ y = B) -> (ph <-> ps))
7	6	biimpd 153	. . . . . 6 |- ((x = A /\ y = B) -> (ph -> ps))
8	7	19.22i2 1039	. . . . 5 |- (E.xE.y(x = A /\ y = B) -> E.xE.y(ph -> ps))
9	5, 8	sylbir 201	. . . 4 |- ((E.x x = A /\ E.y y = B) -> E.xE.y(ph -> ps))
10	2, 4, 9	mp2an 696	. . 3 |- E.xE.y(ph -> ps)
11		19.36v 1298	. . . . 5 |- (E.y(ph -> ps) <-> (A.yph -> ps))
12	11	exbii 1049	. . . 4 |- (E.xE.y(ph -> ps) <-> E.x(A.yph -> ps))
13		19.36v 1298	. . . 4 |- (E.x(A.yph -> ps) <-> (A.xA.yph -> ps))
14	12, 13	bitr 173	. . 3 |- (E.xE.y(ph -> ps) <-> (A.xA.yph -> ps))
15	10, 14	mpbi 189	. 2 |- (A.xA.yph -> ps)
16		vtocl2.4	. . 3 |- ph
17	16	ax-gen 961	. 2 |- A.yph
18	15, 17	mpg 984	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  Vcvv 1807

This theorem is referenced by:  caoprcom 4045  caoprord 4048  ersym 4262

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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