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Theorem 2ralbidv 2511
Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
Hypothesis
Ref Expression
2ralbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
2ralbidv  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  <->  A. 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 2ralbidv
StepHypRef Expression
1 2ralbidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21ralbidv 2487 . 2  |-  ( ph  ->  ( A. y  e.  B  ps  <->  A. y  e.  B  ch )
)
32ralbidv 2487 1  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wral 2465
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 1457  ax-gen 1459  ax-4 1520  ax-17 1536
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-ral 2470
This theorem is referenced by:  cbvral3v  2730  poeq1  4311  soeq1  4327  isoeq1  5815  isoeq2  5816  isoeq3  5817  fnmpoovd  6229  smoeq  6304  xpf1o  6857  tapeq1  7264  elinp  7486  cauappcvgpr  7674  seq3caopr2  10495  addcn2  11331  mulcn2  11333  sgrp1  12834  ismhm  12872  issubm  12882  isnsg  13091  nmznsg  13102  iscmn  13127  ring1  13294  issubrg3  13431  islmod  13444  lmodlema  13445  lsssetm  13509  islssmd  13512  islidlm  13632  ispsmet  14063  ismet  14084  isxmet  14085  addcncntoplem  14291  elcncf  14300
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