Theorem vtoclb 1381

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

Hypotheses

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

Assertion

Ref	Expression
vtoclb	|- (ch <-> th)

Distinct variable group(s):   x,A   ch,x   th,x

Proof of Theorem vtoclb

Step	Hyp	Ref	Expression
1		vtoclb.1	. 2 |- A e. V
2		vtoclb.2	. . 3 |- (x = A -> (ph <-> ch))
3		vtoclb.3	. . 3 |- (x = A -> (ps <-> th))
4	2, 3	bibi12d 477	. 2 |- (x = A -> ((ph <-> ps) <-> (ch <-> th)))
5		vtoclb.4	. 2 |- (ph <-> ps)
6	1, 4, 5	vtocl 1378	1 |- (ch <-> th)

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  eqvinc 1407  alexeq 1409  elpw 1801

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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