Theorem r2ex 2901

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.)

Assertion

Ref	Expression
r2ex	|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Distinct variable groups:   x, y   y, A

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

Proof of Theorem r2ex

Step	Hyp	Ref	Expression
1		r2al 2783	. 2 |- ( A. x e. A A. y e. B -. ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> -. ph ) )
2	1	r2exlem 2899	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 189   /\ wa 376   E.wex 1671   e. wcel 1904   E.wrex 2757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-ral 2761  df-rex 2762

This theorem is referenced by:  reean  2943  rnoprab2  6399  elrnmpt2res  6429  oeeu  7322  omxpenlem  7691  axcnre  9606  hash2prb  12674  pmtrrn2  17179  fsumvma  24220  usgrarnedg  25190  spanuni  27278  5oalem7  27394  3oalem3  27398  elfuns  30753  ellines  30990  dalem20  33329  diblsmopel  34810  iunrelexpuztr  36382  upgredg  39389  umgredg  39390  usgedgimp  40223  usgedgimpALT  40226