Theorem ceqsralv 2712

Description: Restricted quantifier version of ceqsalv 2711. (Contributed by NM, 21-Jun-2013.)

Hypothesis

Ref	Expression
ceqsralv.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

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

Allowed substitution hint:   ph( x)

Proof of Theorem ceqsralv

Step	Hyp	Ref	Expression
1		nfv 1508	. 2 |- F/ x ps
2		ceqsralv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	ax-gen 1425	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		ceqsralt 2708	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )
5	1, 3, 4	mp3an12 1305	1 |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   F/wnf 1436   e. wcel 1480   A.wral 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-v 2683

This theorem is referenced by:  eqreu  2871  sqrt2irr  11829