Theorem r19.29ffa 29320

Description: A commonly used pattern based on r19.29 3072, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)

Hypothesis

Ref	Expression
r19.29ffa.3	|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )

Assertion

Ref	Expression
r19.29ffa	|- ( ( ph /\ E. x e. A E. y e. B ps ) -> ch )

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

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

Proof of Theorem r19.29ffa

Step	Hyp	Ref	Expression
1		r19.29ffa.3	. . . . . . 7 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
2	1	ex 450	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps -> ch ) )
3	2	ralrimiva 2966	. . . . 5 |- ( ( ph /\ x e. A ) -> A. y e. B ( ps -> ch ) )
4	3	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. A A. y e. B ( ps -> ch ) )
5	4	adantr 481	. . 3 |- ( ( ph /\ E. x e. A E. y e. B ps ) -> A. x e. A A. y e. B ( ps -> ch ) )
6		simpr 477	. . 3 |- ( ( ph /\ E. x e. A E. y e. B ps ) -> E. x e. A E. y e. B ps )
7	5, 6	r19.29d2r 3080	. 2 |- ( ( ph /\ E. x e. A E. y e. B ps ) -> E. x e. A E. y e. B ( ( ps -> ch ) /\ ps ) )
8		pm3.35 611	. . . . 5 |- ( ( ps /\ ( ps -> ch ) ) -> ch )
9	8	ancoms 469	. . . 4 |- ( ( ( ps -> ch ) /\ ps ) -> ch )
10	9	rexlimivw 3029	. . 3 |- ( E. y e. B ( ( ps -> ch ) /\ ps ) -> ch )
11	10	rexlimivw 3029	. 2 |- ( E. x e. A E. y e. B ( ( ps -> ch ) /\ ps ) -> ch )
12	7, 11	syl 17	1 |- ( ( ph /\ E. x e. A E. y e. B ps ) -> ch )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  reprsuc  30693