Theorem r2exf 3060

Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3059. (Revised by Wolf Lammen, 10-Jan-2020.)

Hypothesis

Ref	Expression
r2exf.1	|- F/_ y A

Assertion

Ref	Expression
r2exf	|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Distinct variable group:   x, y

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

Proof of Theorem r2exf

Step	Hyp	Ref	Expression
1		r2exf.1	. . 3 |- F/_ y A
2	1	r2alf 2938	. 2 |- ( A. x e. A A. y e. B -. ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> -. ph ) )
3	2	r2exlem 3059	1 |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   F/_wnfc 2751   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918

This theorem is referenced by:  rexcomf  3097