Theorem ceqsralv 2887

Description: Restricted quantifier version of ceqsalv 2886. (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 1619	. 2 |- F/xps
2		ceqsralv.2	. . 3 |- (x = A -> (ph <-> ps))
3	2	ax-gen 1546	. 2 |- A.x(x = A -> (ph <-> ps))
4		ceqsralt 2883	. 2 |- (( F/xps /\ A.x(x = A -> (ph <-> ps)) /\ A e. B) -> (A.x e. B (x = A -> ph) <-> ps))
5	1, 3, 4	mp3an12 1267	1 |- (A e. B -> (A.x e. B (x = A -> ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-v 2862

This theorem is referenced by:  eqreu  3029