Theorem reximdv2 2570

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)

Hypothesis

Ref	Expression
reximdv2.1	|- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ch ) ) )

Assertion

Ref	Expression
reximdv2	|- ( ph -> ( E. x e. A ps -> E. x e. B ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)   B( x)

Proof of Theorem reximdv2

Step	Hyp	Ref	Expression
1		reximdv2.1	. . 3 |- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ch ) ) )
2	1	eximdv 1874	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) -> E. x ( x e. B /\ ch ) ) )
3		df-rex 2455	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
4		df-rex 2455	. 2 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
5	2, 3, 4	3imtr4g 204	1 |- ( ph -> ( E. x e. A ps -> E. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1486   e. wcel 2142   E.wrex 2450

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 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-4 1504  ax-17 1520  ax-ial 1528

This theorem depends on definitions:  df-bi 116  df-rex 2455

This theorem is referenced by:  reximssdv  2575  ssrexv  3213  ssimaex  5561  ico0  10222  ioc0  10223  r19.2uz  10961  trilpolemlt1  14158