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Theorem rexralbidv 2367
Description: Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
Hypothesis
Ref Expression
2ralbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rexralbidv  |-  ( ph  ->  ( E. x  e.  A  A. y  e.  B  ps  <->  E. x  e.  A  A. y  e.  B  ch )
)
Distinct variable groups:    ph, x    ph, y
Allowed substitution hints:    ps( x, y)    ch( x, y)    A( x, y)    B( x, y)

Proof of Theorem rexralbidv
StepHypRef Expression
1 2ralbidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21ralbidv 2343 . 2  |-  ( ph  ->  ( A. y  e.  B  ps  <->  A. y  e.  B  ch )
)
32rexbidv 2344 1  |-  ( ph  ->  ( E. x  e.  A  A. y  e.  B  ps  <->  E. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wral 2323   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-ral 2328  df-rex 2329
This theorem is referenced by:  caucvgpr  6838  caucvgprpr  6868  caucvgsrlemgt1  6937  caucvgsrlemoffres  6942  axcaucvglemres  7031  cvg1nlemres  9812  rexfiuz  9816  resqrexlemgt0  9847  resqrexlemoverl  9848  resqrexlemglsq  9849  resqrexlemsqa  9851  resqrexlemex  9852  cau3lem  9941  caubnd2  9944  climi  10039  2clim  10053
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