Theorem vtocl3ga 1854

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 groups:   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 612	. . . . 5 |- (y = B -> ((A e. D -> ps) <-> (A e. D -> ch)))
3		vtocl3ga.3	. . . . . 6 |- (z = C -> (ch <-> th))
4	3	imbi2d 612	. . . . 5 |- (z = C -> ((A e. D -> ch) <-> (A e. D -> th)))
5		vtocl3ga.1	. . . . . . . 8 |- (x = A -> (ph <-> ps))
6	5	imbi2d 612	. . . . . . 7 |- (x = A -> (((y e. R /\ z e. S) -> ph) <-> ((y e. R /\ z e. S) -> ps)))
7		vtocl3ga.4	. . . . . . . 8 |- ((x e. D /\ y e. R /\ z e. S) -> ph)
8	7	3expib 836	. . . . . . 7 |- (x e. D -> ((y e. R /\ z e. S) -> ph))
9	6, 8	vtoclga 1852	. . . . . 6 |- (A e. D -> ((y e. R /\ z e. S) -> ps))
10	9	com12 11	. . . . 5 |- ((y e. R /\ z e. S) -> (A e. D -> ps))
11	2, 4, 10	vtocl2ga 1853	. . . 4 |- ((B e. R /\ C e. S) -> (A e. D -> th))
12	11	ex 373	. . 3 |- (B e. R -> (C e. S -> (A e. D -> th)))
13	12	com3r 35	. 2 |- (A e. D -> (B e. R -> (C e. S -> th)))
14	13	3imp 827	1 |- ((A e. D /\ B e. R /\ C e. S) -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958

This theorem is referenced by:  pocl 2844

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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