Theorem reximia 2589

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)

Hypothesis

Ref	Expression
reximia.1	|- ( x e. A -> ( ph -> ps ) )

Assertion

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

Proof of Theorem reximia

Step	Hyp	Ref	Expression
1		rexim 2588	. 2 |- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph -> E. x e. A ps ) )
2		reximia.1	. 2 |- ( x e. A -> ( ph -> ps ) )
3	1, 2	mprg 2551	1 |- ( E. x e. A ph -> E. x e. A ps )

Syntax hints:   -> wi 4   e. wcel 2164   E.wrex 2473

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  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-ral 2477  df-rex 2478

This theorem is referenced by:  reximi  2591  iunpw  4511  nsmallnqq  7472  1idprl  7650  1idpru  7651  qmulz  9688  zq  9691  caubnd2  11261  sin0pilem1  14916