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Theorem 2rexbiia 2741
Description: Inference adding two 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 2724 . 2  |-  ( x  e.  A  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ps ) )
32rexbiia 2740 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 178    /\ wa 360    e. wcel 1726   E.wrex 2708
This theorem is referenced by:  cnref1o  10612  mdsymlem8  23918  xlt2addrd  24129  elunirnmbfm  24608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-rex 2713
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