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Theorem ralbida 2484
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1  |-  F/ x ph
ralbida.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralbida  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)

Proof of Theorem ralbida
StepHypRef Expression
1 ralbida.1 . . 3  |-  F/ x ph
2 ralbida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.74da 443 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  A  ->  ch ) ) )
41, 3albid 1626 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. x ( x  e.  A  ->  ch ) ) )
5 df-ral 2473 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
6 df-ral 2473 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
74, 5, 63bitr4g 223 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362   F/wnf 1471    e. wcel 2160   A.wral 2468
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-4 1521
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2473
This theorem is referenced by:  ralbidva  2486  ralbid  2488  2ralbida  2511  ralbi  2622  ismkvnex  7183  caucvgsrlemgt1  7824  iswomninnlem  15256
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