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Theorem r2ex 2495
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
StepHypRef Expression
1 nfcv 2317 . 2  |-  F/_ y A
21r2exf 2493 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   E.wex 1490    e. wcel 2146   E.wrex 2454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459
This theorem is referenced by:  reean  2643  rexiunxp  4762  rnoprab2  5949  genprndl  7495  genprndu  7496  genpdisj  7497  prmuloc  7540  mullocpr  7545  axcnre  7855
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