Theorem r19.29vva 2642

Description: A commonly used pattern based on r19.29 2634, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)

Hypotheses

Ref	Expression
r19.29vva.1	|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
r19.29vva.2	|- ( ph -> E. x e. A E. y e. B ps )

Assertion

Ref	Expression
r19.29vva	|- ( ph -> ch )

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

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

Proof of Theorem r19.29vva

Step	Hyp	Ref	Expression
1		r19.29vva.1	. . . . . 6 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
2	1	ex 115	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps -> ch ) )
3	2	ralrimiva 2570	. . . 4 |- ( ( ph /\ x e. A ) -> A. y e. B ( ps -> ch ) )
4	3	ralrimiva 2570	. . 3 |- ( ph -> A. x e. A A. y e. B ( ps -> ch ) )
5		r19.29vva.2	. . 3 |- ( ph -> E. x e. A E. y e. B ps )
6	4, 5	r19.29d2r 2641	. 2 |- ( ph -> E. x e. A E. y e. B ( ( ps -> ch ) /\ ps ) )
7		pm3.35 347	. . . . 5 |- ( ( ps /\ ( ps -> ch ) ) -> ch )
8	7	ancoms 268	. . . 4 |- ( ( ( ps -> ch ) /\ ps ) -> ch )
9	8	rexlimivw 2610	. . 3 |- ( E. y e. B ( ( ps -> ch ) /\ ps ) -> ch )
10	9	rexlimivw 2610	. 2 |- ( E. x e. A E. y e. B ( ( ps -> ch ) /\ ps ) -> ch )
11	6, 10	syl 14	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   A.wral 2475   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by: (None)