Theorem r2exf

Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016) Use r2exlem . (Revised by Wolf Lammen, 10-Jan-2020)

	Ref	Expression
Hypothesis	r2exf.1	|- F/_ y A
Assertion	r2exf	|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		r2exf.1	|- F/_ y A
2	1	r2alf	|- ( A. x e. A A. y e. B -. ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> -. ph ) )
3	2	r2exlem	|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )