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Theorem 2rexbiia 2577
Description: Inference adding 2 restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
Hypothesis
Ref Expression
2rexbiia.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
2rexbiia  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  ps )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x)    B( x, y)

Proof of Theorem 2rexbiia
StepHypRef Expression
1 2rexbiia.1 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ph  <->  ps )
)
21rexbidva 2560 . 2  |-  ( x  e.  A  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ps ) )
32rexbiia 2576 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  cnref1o  10349  mdsymlem8  22990  xlt2addrd  23253  elunirnmbfm  23558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-rex 2549
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