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Theorem 2rexbiia 2357
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 2340 . 2  |-  ( x  e.  A  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ps ) )
32rexbiia 2356 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    /\ wa 101    <-> wb 102    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-rex 2329
This theorem is referenced by:  elq  8654  cnref1o  8680
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