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Theorem ralbida 2684
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 669 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  A  ->  ch ) ) )
41, 3albid 1784 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. x ( x  e.  A  ->  ch ) ) )
5 df-ral 2675 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
6 df-ral 2675 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
74, 5, 63bitr4g 280 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    e. wcel 1721   A.wral 2670
This theorem is referenced by:  ralbidva  2686  ralbid  2688  2ralbida  2709  ralbi  2806  ac6num  8319  neiptopreu  17156  funcnv5mpt  24041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-ral 2675
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