Theorem vtoclf 1377

Description: Implicit substitution of a class for a set variable. This is a generalization of chv2 850.

Hypotheses

Ref	Expression
vtoclf.1	|- (ps -> A.xps)
vtoclf.2	|- A e. V
vtoclf.3	|- (x = A -> (ph <-> ps))
vtoclf.4	|- ph

Assertion

Ref	Expression
vtoclf	|- ps

Distinct variable group(s):   x,A

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. . 3 |- (ps -> A.xps)
2		vtoclf.2	. . . . 5 |- A e. V
3	2	isseti 1352	. . . 4 |- E.x x = A
4		vtoclf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
5	4	biimpd 135	. . . . 5 |- (x = A -> (ph -> ps))
6	5	19.22i 723	. . . 4 |- (E.x x = A -> E.x(ph -> ps))
7	3, 6	ax-mp 6	. . 3 |- E.x(ph -> ps)
8	1, 7	19.36i 758	. 2 |- (A.xph -> ps)
9		vtoclf.4	. 2 |- ph
10	8, 9	mpg 684	1 |- ps

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  vtocl 1378  axrep2 1474  zfcndrep 3760

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-gen 677  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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