Theorem reximd2a 3013

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Hypotheses

Ref	Expression
reximd2a.1	|- F/ x ph
reximd2a.2	|- ( ( ( ph /\ x e. A ) /\ ps ) -> x e. B )
reximd2a.3	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
reximd2a.4	|- ( ph -> E. x e. A ps )

Assertion

Ref	Expression
reximd2a	|- ( ph -> E. x e. B ch )

Proof of Theorem reximd2a

Step	Hyp	Ref	Expression
1		reximd2a.4	. 2 |- ( ph -> E. x e. A ps )
2		reximd2a.1	. . . 4 |- F/ x ph
3		reximd2a.2	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ps ) -> x e. B )
4		reximd2a.3	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
5	3, 4	jca 554	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ( x e. B /\ ch ) )
6	5	expl 648	. . . 4 |- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ch ) ) )
7	2, 6	eximd 2085	. . 3 |- ( ph -> ( E. x ( x e. A /\ ps ) -> E. x ( x e. B /\ ch ) ) )
8		df-rex 2918	. . 3 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
9		df-rex 2918	. . 3 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
10	7, 8, 9	3imtr4g 285	. 2 |- ( ph -> ( E. x e. A ps -> E. x e. B ch ) )
11	1, 10	mpd 15	1 |- ( ph -> E. x e. B ch )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   F/wnf 1708   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  locfinreflem  29907