Theorem ceqsralv 2808

Description: Restricted quantifier version of ceqsalv 2807. (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 1552	. 2 |- F/ x ps
2		ceqsralv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	ax-gen 1473	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		ceqsralt 2804	. 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 1340	1 |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   e. wcel 2178   A.wral 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-v 2778

This theorem is referenced by:  eqreu  2972  sqrt2irr  12599