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Theorem hbbid 1537
Description: Deduction form of bound-variable hypothesis builder hbbi 1510. (Contributed by NM, 1-Jan-2002.)
Hypotheses
Ref Expression
hbbid.1  |-  ( ph  ->  A. x ph )
hbbid.2  |-  ( ph  ->  ( ps  ->  A. x ps ) )
hbbid.3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
Assertion
Ref Expression
hbbid  |-  ( ph  ->  ( ( ps  <->  ch )  ->  A. x ( ps  <->  ch ) ) )

Proof of Theorem hbbid
StepHypRef Expression
1 hbbid.1 . . . 4  |-  ( ph  ->  A. x ph )
2 hbbid.2 . . . 4  |-  ( ph  ->  ( ps  ->  A. x ps ) )
3 hbbid.3 . . . 4  |-  ( ph  ->  ( ch  ->  A. x ch ) )
41, 2, 3hbimd 1535 . . 3  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )
51, 3, 2hbimd 1535 . . 3  |-  ( ph  ->  ( ( ch  ->  ps )  ->  A. x
( ch  ->  ps ) ) )
64, 5anim12d 331 . 2  |-  ( ph  ->  ( ( ( ps 
->  ch )  /\  ( ch  ->  ps ) )  ->  ( A. x
( ps  ->  ch )  /\  A. x ( ch  ->  ps )
) ) )
7 dfbi2 383 . 2  |-  ( ( ps  <->  ch )  <->  ( ( ps  ->  ch )  /\  ( ch  ->  ps )
) )
8 albiim 1446 . 2  |-  ( A. x ( ps  <->  ch )  <->  ( A. x ( ps 
->  ch )  /\  A. x ( ch  ->  ps ) ) )
96, 7, 83imtr4g 204 1  |-  ( ph  ->  ( ( ps  <->  ch )  ->  A. x ( ps  <->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312
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 1406  ax-gen 1408  ax-4 1470  ax-i5r 1498
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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