Theorem vtocl2gf 1385

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 group(s):   x,A   y,A   y,B

Proof of Theorem vtocl2gf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (z e. B -> A.y z e. B)
2		ax-17 925	. . . . . 6 |- (A e. V -> A.y A e. V)
3		vtocl2gf.2	. . . . . 6 |- (ch -> A.ych)
4	2, 3	hbim 702	. . . . 5 |- ((A e. V -> ch) -> A.y(A e. V -> ch))
5		vtocl2gf.4	. . . . . 6 |- (y = B -> (ps <-> ch))
6	5	imbi2d 464	. . . . 5 |- (y = B -> ((A e. V -> ps) <-> (A e. V -> ch)))
7		ax-17 925	. . . . . 6 |- (z e. A -> A.x z e. A)
8		vtocl2gf.1	. . . . . 6 |- (ps -> A.xps)
9		vtocl2gf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
10		vtocl2gf.5	. . . . . 6 |- ph
11	7, 8, 9, 10	vtoclgf 1382	. . . . 5 |- (A e. V -> ps)
12	1, 4, 6, 11	vtoclgf 1382	. . . 4 |- (B e. D -> (A e. V -> ch))
13	12	com12 13	. . 3 |- (A e. V -> (B e. D -> ch))
14	13	imp 277	. 2 |- ((A e. V /\ B e. D) -> ch)
15		elisset 1354	. 2 |- (A e. C -> A e. V)
16	14, 15	sylan 343	1 |- ((A e. C /\ B e. D) -> ch)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  vtocl2g 1386

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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