Theorem reximddv 2569

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

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

Assertion

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

Distinct variable group:   ph, x

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

Proof of Theorem reximddv

Step	Hyp	Ref	Expression
1		reximddva.2	. 2 |- ( ph -> E. x e. A ps )
2		reximddva.1	. . . 4 |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch )
3	2	expr 373	. . 3 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
4	3	reximdva 2568	. 2 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )
5	1, 4	mpd 13	1 |- ( ph -> E. x e. A ch )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   E.wrex 2445

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  reximddv2  2571