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Theorem 2rexbidva 2513
Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
Hypothesis
Ref Expression
2ralbidva.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  <->  ch )
)
Assertion
Ref Expression
2rexbidva  |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps  <->  E. x  e.  A  E. y  e.  B  ch )
)
Distinct variable groups:    x, y, ph    y, A
Allowed substitution hints:    ps( x, y)    ch( x, y)    A( x)    B( x, y)

Proof of Theorem 2rexbidva
StepHypRef Expression
1 2ralbidva.1 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  <->  ch )
)
21anassrs 400 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps 
<->  ch ) )
32rexbidva 2487 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( E. y  e.  B  ps 
<->  E. y  e.  B  ch ) )
43rexbidva 2487 1  |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps  <->  E. x  e.  A  E. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2160   E.wrex 2469
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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-rex 2474
This theorem is referenced by:  pythagtriplem2  12298  pythagtrip  12315
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