Theorem vtoclri 1850

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 1689	. 2 |- (x e. B -> ph)
4	1, 3	vtoclga 1843	1 |- (A e. B -> ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  A.wral 1637

This theorem is referenced by:  omsdomnn 4509  arch 6018  discrlem 6589  climabslem 7084  climcau 7092  ivthlem2 7217  ivthlem8 7223  ivthlem8OLD 7232  hlimcaui 9027  ghomgrpilem1 10290

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803

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