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Theorem 2rexbidva 2364
Description: Formula-building rule for restricted existential quantifiers (deduction rule). (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 386 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps 
<->  ch ) )
32rexbidva 2340 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( E. y  e.  B  ps 
<->  E. y  e.  B  ch ) )
43rexbidva 2340 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 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: (None)
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