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Theorem ralbid 2412
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
ralbid.1  |-  F/ x ph
ralbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
ralbid  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)

Proof of Theorem ralbid
StepHypRef Expression
1 ralbid.1 . 2  |-  F/ x ph
2 ralbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
32adantr 274 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
41, 3ralbida 2408 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1421    e. wcel 1465   A.wral 2393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-4 1472
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-ral 2398
This theorem is referenced by:  ralbidv  2414  sbcralt  2957  riota5f  5722  mkvprop  7000  lble  8673  ellimc3apf  12725
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