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Theorem ralbida 2841
Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.) Shortened after introduction of hbralbida 2836. (Revised by Wolf Lammen, 5-Dec-2019.)
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
21nfri 1813 . 2  |-  ( ph  ->  A. x ph )
3 ralbida.2 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
42, 3hbralbida 2836 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 369   F/wnf 1590    e. wcel 1758   A.wral 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-ral 2804
This theorem is referenced by:  ralbid  2843  ralbidvaOLD  2845  2ralbida  2849  ralbi  2959  ac6num  8760  neiptopreu  18870  istrkg2ld  23056  funcnv5mpt  26140
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