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Theorem hband 1394
Description: Deduction form of bound-variable hypothesis builder hban 1455. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
hband.1  |-  ( ph  ->  ( ps  ->  A. x ps ) )
hband.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
Assertion
Ref Expression
hband  |-  ( ph  ->  ( ( ps  /\  ch )  ->  A. x
( ps  /\  ch ) ) )

Proof of Theorem hband
StepHypRef Expression
1 hband.1 . . 3  |-  ( ph  ->  ( ps  ->  A. x ps ) )
2 hband.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
31, 2anim12d 322 . 2  |-  ( ph  ->  ( ( ps  /\  ch )  ->  ( A. x ps  /\  A. x ch ) ) )
4 19.26 1386 . 2  |-  ( A. x ( ps  /\  ch )  <->  ( A. x ps  /\  A. x ch ) )
53, 4syl6ibr 155 1  |-  ( ph  ->  ( ( ps  /\  ch )  ->  A. x
( ps  /\  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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