Theorem vtoclb 1848

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 groups:   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 631	. 2 |- (x = A -> ((ph <-> ps) <-> (ch <-> th)))
5		vtoclb.4	. 2 |- (ph <-> ps)
6	1, 4, 5	vtocl 1845	1 |- (ch <-> th)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814

This theorem is referenced by:  eqvinc 1886  alexeq 1888  elpw 2408

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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