Theorem vtocl2ga 1388

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 group(s):   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	. . . . 5 |- (y = B -> (ps <-> ch))
2	1	imbi2d 464	. . . 4 |- (y = B -> ((A e. C -> ps) <-> (A e. C -> ch)))
3		vtocl2ga.1	. . . . . . 7 |- (x = A -> (ph <-> ps))
4	3	imbi2d 464	. . . . . 6 |- (x = A -> ((y e. D -> ph) <-> (y e. D -> ps)))
5		vtocl2ga.3	. . . . . . 7 |- ((x e. C /\ y e. D) -> ph)
6	5	exp 291	. . . . . 6 |- (x e. C -> (y e. D -> ph))
7	4, 6	vtoclga 1387	. . . . 5 |- (A e. C -> (y e. D -> ps))
8	7	com12 13	. . . 4 |- (y e. D -> (A e. C -> ps))
9	2, 8	vtoclga 1387	. . 3 |- (B e. D -> (A e. C -> ch))
10	9	com12 13	. 2 |- (A e. C -> (B e. D -> ch))
11	10	imp 277	1 |- ((A e. C /\ B e. D) -> ch)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092

This theorem is referenced by:  vtocl3ga 1389  solin 2145  f1fveq 2918  caoprcl 3066  caoprcan 3069  ltpiord 3809  genpv 3896  expcllem 4682

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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