Theorem vtocl3ga 1389

Description: Implicit substitution of 3 classes for 3 set variables.

Hypotheses

Ref	Expression
vtocl3ga.1	|- (x = A -> (ph <-> ps))
vtocl3ga.2	|- (y = B -> (ps <-> ch))
vtocl3ga.3	|- (z = C -> (ch <-> th))
vtocl3ga.4	|- ((x e. D /\ y e. R /\ z e. S) -> ph)

Assertion

Ref	Expression
vtocl3ga	|- ((A e. D /\ B e. R /\ C e. S) -> th)

Distinct variable group(s):   x,y,z,A   y,B,z   z,C   x,D,y,z   x,R,y,z   x,S,y,z   ps,x   ch,y   th,z

Proof of Theorem vtocl3ga

Step	Hyp	Ref	Expression
1		vtocl3ga.2	. . . . . 6 |- (y = B -> (ps <-> ch))
2	1	imbi2d 464	. . . . 5 |- (y = B -> ((A e. D -> ps) <-> (A e. D -> ch)))
3		vtocl3ga.3	. . . . . 6 |- (z = C -> (ch <-> th))
4	3	imbi2d 464	. . . . 5 |- (z = C -> ((A e. D -> ch) <-> (A e. D -> th)))
5		vtocl3ga.1	. . . . . . . 8 |- (x = A -> (ph <-> ps))
6	5	imbi2d 464	. . . . . . 7 |- (x = A -> (((y e. R /\ z e. S) -> ph) <-> ((y e. R /\ z e. S) -> ps)))
7		vtocl3ga.4	. . . . . . . . 9 |- ((x e. D /\ y e. R /\ z e. S) -> ph)
8	7	3exp 611	. . . . . . . 8 |- (x e. D -> (y e. R -> (z e. S -> ph)))
9	8	imp3a 279	. . . . . . 7 |- (x e. D -> ((y e. R /\ z e. S) -> ph))
10	6, 9	vtoclga 1387	. . . . . 6 |- (A e. D -> ((y e. R /\ z e. S) -> ps))
11	10	com12 13	. . . . 5 |- ((y e. R /\ z e. S) -> (A e. D -> ps))
12	2, 4, 11	vtocl2ga 1388	. . . 4 |- ((B e. R /\ C e. S) -> (A e. D -> th))
13	12	exp 291	. . 3 |- (B e. R -> (C e. S -> (A e. D -> th)))
14	13	com3r 35	. 2 |- (A e. D -> (B e. R -> (C e. S -> th)))
15	14	3imp 608	1 |- ((A e. D /\ B e. R /\ C e. S) -> th)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092

This theorem is referenced by:  pocl 2132

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-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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