Theorem rexlim2d 39857

Description: Inference removing two restricted quantifiers. Same as rexlimdvv 3037, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
rexlim2d.x	|- F/ x ph
rexlim2d.y	|- F/ y ph
rexlim2d.3	|- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) )

Assertion

Ref	Expression
rexlim2d	|- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

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

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

Proof of Theorem rexlim2d

Step	Hyp	Ref	Expression
1		rexlim2d.x	. 2 |- F/ x ph
2		nfv 1843	. 2 |- F/ x ch
3		rexlim2d.y	. . . . 5 |- F/ y ph
4		nfv 1843	. . . . 5 |- F/ y x e. A
5	3, 4	nfan 1828	. . . 4 |- F/ y ( ph /\ x e. A )
6		nfv 1843	. . . 4 |- F/ y ch
7		rexlim2d.3	. . . . 5 |- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) )
8	7	expdimp 453	. . . 4 |- ( ( ph /\ x e. A ) -> ( y e. B -> ( ps -> ch ) ) )
9	5, 6, 8	rexlimd 3026	. . 3 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> ch ) )
10	9	ex 450	. 2 |- ( ph -> ( x e. A -> ( E. y e. B ps -> ch ) ) )
11	1, 2, 10	rexlimd 3026	1 |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 384   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-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  fourierdlem48  40371