Theorem reximi2 2646

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 1653	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ( x e. B /\ ps ) )
3		df-rex 2534	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2534	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-rex 2534

This theorem is used by:  btwnz  9765  ioo0  10694  ballotfilemfc0  13232  ballotfilemfcc  13233  lgsquadlem2  16197