Theorem reximddv2 2571

Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
reximddv2.1	|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
reximddv2.2	|- ( ph -> E. x e. A E. y e. B ps )

Assertion

Ref	Expression
reximddv2	|- ( ph -> E. x e. A E. y e. B ch )

Distinct variable groups:   y, A   ph, x, y

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

Proof of Theorem reximddv2

Step	Hyp	Ref	Expression
1		reximddv2.1	. . . . 5 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
2	1	ex 114	. . . 4 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps -> ch ) )
3	2	reximdva 2568	. . 3 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> E. y e. B ch ) )
4	3	impr 377	. 2 |- ( ( ph /\ ( x e. A /\ E. y e. B ps ) ) -> E. y e. B ch )
5		reximddv2.2	. 2 |- ( ph -> E. x e. A E. y e. B ps )
6	4, 5	reximddv 2569	1 |- ( ph -> E. x e. A E. y e. B 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: (None)