Theorem vtoclf 1841

Description: Implicit substitution of a class for a set variable. This is a generalization of chvar 1167.

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 1815	. . . 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 1040	. . . 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 1079	. 2 |- (A.xph -> ps)
9		vtoclf.4	. 2 |- ph
10	8, 9	mpg 986	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  vtocl 1842

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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