Theorem ralimiaa 2556

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)

Hypothesis

Ref	Expression
ralimiaa.1	|- ( ( x e. A /\ ph ) -> ps )

Assertion

Ref	Expression
ralimiaa	|- ( A. x e. A ph -> A. x e. A ps )

Proof of Theorem ralimiaa

Step	Hyp	Ref	Expression
1		ralimiaa.1	. . 3 |- ( ( x e. A /\ ph ) -> ps )
2	1	ex 115	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	ralimia 2555	1 |- ( A. x e. A ph -> A. x e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   A.wral 2472

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 1458  ax-gen 1460

This theorem depends on definitions:  df-bi 117  df-ral 2477

This theorem is referenced by:  ralrnmpt  5700  rexrnmpt  5701  acexmidlem2  5915  mptelixpg  6788  trirec0  15534