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Theorem r2ex 2743
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 2572 . 2  |-  F/_ y A
21r2exf 2741 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:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   E.wrex 2706
This theorem is referenced by:  reean  2874  rnoprab2  6157  oeeu  6846  omxpenlem  7209  axcnre  9039  fsumvma  20997  usgrarnedg  21404  spanuni  23046  5oalem7  23162  3oalem3  23166  elfuns  25760  ellines  26086  dalem20  30490  diblsmopel  31969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711
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