Theorem r2ex 2429

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)

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		nfcv 2255	. 2 |- F/_ y A
2	1	r2exf 2427	1 |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1451   e. wcel 1463   E.wrex 2391

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396

This theorem is referenced by:  reean  2573  rexiunxp  4641  rnoprab2  5809  genprndl  7277  genprndu  7278  genpdisj  7279  prmuloc  7322  mullocpr  7327  axcnre  7616