Theorem vtocl2ga 2353

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 674	. . 3 |- (y = B -> ((A e. C -> ps) <-> (A e. C -> ch)))
3		vtocl2ga.1	. . . . . 6 |- (x = A -> (ph <-> ps))
4	3	imbi2d 674	. . . . 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 402	. . . . 5 |- (x e. C -> (y e. D -> ph))
7	4, 6	vtoclga 2352	. . . 4 |- (A e. C -> (y e. D -> ps))
8	7	com12 14	. . 3 |- (y e. D -> (A e. C -> ps))
9	2, 8	vtoclga 2352	. 2 |- (B e. D -> (A e. C -> ch))
10	9	impcom 378	1 |- ((A e. C /\ B e. D) -> ch)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300

This theorem is referenced by:  vtocl3ga 2354  solin 3612  f1fveq 4852  caoprcl 4985  caoprcan 4988  ltpiord 6167  genpv 6254  expcllem 7818  isgrp2i 9360  issubgilem 9430  htthlem2 9968  opsqrlem4 11714  cptarc 15242

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-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  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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