Theorem vtocl2ga 1844

Description: Implicit substitution of 2 classes for 2 set variables.

Hypotheses

Ref	Expression
vtocl2ga.1	|- (x = A -> (ph <-> ps))
vtocl2ga.2	|- (y = B -> (ps <-> ch))
vtocl2ga.3	|- ((x e. C /\ y e. D) -> ph)

Assertion

Ref	Expression
vtocl2ga	|- ((A e. C /\ B e. D) -> ch)

Distinct variable groups:   x,y,A   y,B   x,C,y   x,D,y   ps,x   ch,y

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		vtocl2ga.2	. . . 4 |- (y = B -> (ps <-> ch))
2	1	imbi2d 610	. . 3 |- (y = B -> ((A e. C -> ps) <-> (A e. C -> ch)))
3		vtocl2ga.1	. . . . . 6 |- (x = A -> (ph <-> ps))
4	3	imbi2d 610	. . . . 5 |- (x = A -> ((y e. D -> ph) <-> (y e. D -> ps)))
5		vtocl2ga.3	. . . . . 6 |- ((x e. C /\ y e. D) -> ph)
6	5	ex 373	. . . . 5 |- (x e. C -> (y e. D -> ph))
7	4, 6	vtoclga 1843	. . . 4 |- (A e. C -> (y e. D -> ps))
8	7	com12 11	. . 3 |- (y e. D -> (A e. C -> ps))
9	2, 8	vtoclga 1843	. 2 |- (B e. D -> (A e. C -> ch))
10	9	impcom 351	1 |- ((A e. C /\ B e. D) -> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955

This theorem is referenced by:  vtocl3ga 1845  solin 2848  f1fveq 3861  caoprcl 4038  caoprcan 4041  ltpiord 4987  genpv 5074  expcllem 6507  isgrp2i 8011  issubgilem 8058  htthlem2 8551

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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