Theorem vtocl2gf 1840

Description: Implicit substitution of a class for a set variable.

Hypotheses

Ref	Expression
vtocl2gf.1	|- (ps -> A.xps)
vtocl2gf.2	|- (ch -> A.ych)
vtocl2gf.3	|- (x = A -> (ph <-> ps))
vtocl2gf.4	|- (y = B -> (ps <-> ch))
vtocl2gf.5	|- ph

Assertion

Ref	Expression
vtocl2gf	|- ((A e. C /\ B e. D) -> ch)

Distinct variable groups:   x,A   y,A   y,B

Proof of Theorem vtocl2gf

Step	Hyp	Ref	Expression
1		ax-17 968	. . . 4 |- (z e. B -> A.y z e. B)
2		ax-17 968	. . . . 5 |- (A e. V -> A.y A e. V)
3		vtocl2gf.2	. . . . 5 |- (ch -> A.ych)
4	2, 3	hbim 1004	. . . 4 |- ((A e. V -> ch) -> A.y(A e. V -> ch))
5		vtocl2gf.4	. . . . 5 |- (y = B -> (ps <-> ch))
6	5	imbi2d 610	. . . 4 |- (y = B -> ((A e. V -> ps) <-> (A e. V -> ch)))
7		ax-17 968	. . . . 5 |- (z e. A -> A.x z e. A)
8		vtocl2gf.1	. . . . 5 |- (ps -> A.xps)
9		vtocl2gf.3	. . . . 5 |- (x = A -> (ph <-> ps))
10		vtocl2gf.5	. . . . 5 |- ph
11	7, 8, 9, 10	vtoclgf 1837	. . . 4 |- (A e. V -> ps)
12	1, 4, 6, 11	vtoclgf 1837	. . 3 |- (B e. D -> (A e. V -> ch))
13	12	impcom 351	. 2 |- ((A e. V /\ B e. D) -> ch)
14		elisset 1808	. 2 |- (A e. C -> A e. V)
15	13, 14	sylan 448	1 |- ((A e. C /\ B e. D) -> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802

This theorem is referenced by:  vtocl2g 1841

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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