Theorem rexlim2d

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

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

Proof

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