Theorem r2exf 2927

Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 2926. (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 2779	. 2 |- ( A. x e. A A. y e. B -. ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> -. ph ) )
3	2	r2exlem 2926	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 184   /\ wa 367   E.wex 1633   e. wcel 1842   F/_wnfc 2550   E.wrex 2754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759

This theorem is referenced by:  r2exOLD  2930  rexcomf  2966