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Theorem ralbidv2 2382
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
Hypothesis
Ref Expression
ralbidv2.1  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch ) ) )
Assertion
Ref Expression
ralbidv2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem ralbidv2
StepHypRef Expression
1 ralbidv2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch ) ) )
21albidv 1752 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. x ( x  e.  B  ->  ch ) ) )
3 df-ral 2364 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
4 df-ral 2364 . 2  |-  ( A. x  e.  B  ch  <->  A. x ( x  e.  B  ->  ch )
)
52, 3, 43bitr4g 221 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    e. wcel 1438   A.wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-ral 2364
This theorem is referenced by:  ralss  3085  dfsmo2  6034  raluz  9035  isprm3  11182
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