Theorem reximdd 39344

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

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

Assertion

Ref	Expression
reximdd	|- ( ph -> E. x e. A ch )

Proof of Theorem reximdd

Step	Hyp	Ref	Expression
1		reximdd.3	. 2 |- ( ph -> E. x e. A ps )
2		reximdd.1	. . 3 |- F/ x ph
3		reximdd.2	. . . 4 |- ( ( ph /\ x e. A /\ ps ) -> ch )
4	3	3exp 1264	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
5	2, 4	reximdai 3012	. 2 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )
6	1, 5	mpd 15	1 |- ( ph -> E. x e. A ch )

Syntax hints:   -> wi 4   /\ w3a 1037   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-3an 1039  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  xlimmnfvlem2  40059  xlimmnfv  40060  xlimpnfvlem2  40063  xlimpnfv  40064