Theorem reximi2 2593

Description: Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)

Hypothesis

Ref	Expression
reximi2.1	|- ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )

Assertion

Ref	Expression
reximi2	|- ( E. x e. A ph -> E. x e. B ps )

Proof of Theorem reximi2

Step	Hyp	Ref	Expression
1		reximi2.1	. . 3 |- ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) )
2	1	eximi 1614	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ps ) )
3		df-rex 2481	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2481	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
5	2, 3, 4	3imtr4i 201	1 |- ( E. x e. A ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1506   e. wcel 2167   E.wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by:  btwnz  9445  ioo0  10349  lgsquadlem2  15319