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Theorem rexralbidv 2404
Description: Formula-building rule for restricted quantifiers (deduction form). (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 2380 . 2  |-  ( ph  ->  ( A. y  e.  B  ps  <->  A. y  e.  B  ch )
)
32rexbidv 2381 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 103   A.wral 2359   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364  df-rex 2365
This theorem is referenced by:  caucvgpr  7220  caucvgprpr  7250  caucvgsrlemgt1  7319  caucvgsrlemoffres  7324  axcaucvglemres  7413  cvg1nlemres  10383  rexfiuz  10387  resqrexlemgt0  10418  resqrexlemoverl  10419  resqrexlemglsq  10420  resqrexlemsqa  10422  resqrexlemex  10423  cau3lem  10512  caubnd2  10515  climi  10639  2clim  10653
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