Theorem vtoclri 1855

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

Hypotheses

Ref	Expression
vtoclri.1	|- (x = A -> (ph <-> ps))
vtoclri.2	|- A.x e. B ph

Assertion

Ref	Expression
vtoclri	|- (A e. B -> ps)

Distinct variable groups:   x,A   x,B   ps,x

Proof of Theorem vtoclri

Step	Hyp	Ref	Expression
1		vtoclri.1	. 2 |- (x = A -> (ph <-> ps))
2		vtoclri.2	. . 3 |- A.x e. B ph
3	2	rspec 1694	. 2 |- (x e. B -> ph)
4	1, 3	vtoclga 1848	1 |- (A e. B -> ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  A.wral 1642

This theorem is referenced by:  omsdomnn 4515  arch 6026  discrlem 6597  climabslem 7092  climcau 7100  ivthlem2 7225  ivthlem8 7231  ivthlem8OLD 7240  hlimcaui 9045  ghomgrpilem1 10319

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808

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