Theorem reximi2 2457

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 1531	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ps ) )
3		df-rex 2354	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2354	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
5	2, 3, 4	3imtr4i 199	1 |- ( E. x e. A ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-rex 2354

This theorem is referenced by:  btwnz  8466  ioo0  9268