Theorem ralimiaa 2532

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 114	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	ralimia 2531	1 |- ( A. x e. A ph -> A. x e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448

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 1440  ax-gen 1442

This theorem depends on definitions:  df-bi 116  df-ral 2453

This theorem is referenced by:  ralrnmpt  5638  rexrnmpt  5639  acexmidlem2  5850  mptelixpg  6712  trirec0  14076