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Theorem ralbid2 2461
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Hypotheses
Ref Expression
ralbid2.nf  |-  F/ x ph
ralbid2.1  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch ) ) )
Assertion
Ref Expression
ralbid2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)

Proof of Theorem ralbid2
StepHypRef Expression
1 ralbid2.nf . . 3  |-  F/ x ph
2 ralbid2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch ) ) )
31, 2albid 1595 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. x ( x  e.  B  ->  ch ) ) )
4 df-ral 2440 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
5 df-ral 2440 . 2  |-  ( A. x  e.  B  ch  <->  A. x ( x  e.  B  ->  ch )
)
63, 4, 53bitr4g 222 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1333   F/wnf 1440    e. wcel 2128   A.wral 2435
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 1427  ax-gen 1429  ax-4 1490
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440
This theorem is referenced by:  strcollnft  13630
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