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Theorem ralbida 2887
Description: Formula-building rule for restricted universal quantifier (deduction rule). (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 685 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  A  ->  ch ) ) )
41, 3albid 1890 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. x ( x  e.  A  ->  ch ) ) )
5 df-ral 2809 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
6 df-ral 2809 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
74, 5, 63bitr4g 288 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396   F/wnf 1621    e. wcel 1823   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-ral 2809
This theorem is referenced by:  ralbid  2888  ralbidvaOLD  2891  2ralbida  2895  ralbi  2985  ac6num  8850  neiptopreu  19801  istrkg2ld  24056  funcnv5mpt  27738
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