Theorem vtoclf 1838

Description: Implicit substitution of a class for a set variable. This is a generalization of chvar 1166.

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:   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 1812	. . . 4 |- E.x x = A
4		vtoclf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
5	4	biimpd 153	. . . . 5 |- (x = A -> (ph -> ps))
6	5	19.22i 1039	. . . 4 |- (E.x x = A -> E.x(ph -> ps))
7	3, 6	ax-mp 7	. . 3 |- E.x(ph -> ps)
8	1, 7	19.36i 1078	. 2 |- (A.xph -> ps)
9		vtoclf.4	. 2 |- ph
10	8, 9	mpg 985	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808

This theorem is referenced by:  vtocl 1839

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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