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Theorem hb3and 1420
Description: Deduction form of bound-variable hypothesis builder hb3an 1483. (Contributed by NM, 17-Feb-2013.)
Hypotheses
Ref Expression
hb3and.1  |-  ( ph  ->  ( ps  ->  A. x ps ) )
hb3and.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
hb3and.3  |-  ( ph  ->  ( th  ->  A. x th ) )
Assertion
Ref Expression
hb3and  |-  ( ph  ->  ( ( ps  /\  ch  /\  th )  ->  A. x ( ps  /\  ch  /\  th ) ) )

Proof of Theorem hb3and
StepHypRef Expression
1 hb3and.1 . . 3  |-  ( ph  ->  ( ps  ->  A. x ps ) )
2 hb3and.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
3 hb3and.3 . . 3  |-  ( ph  ->  ( th  ->  A. x th ) )
41, 2, 33anim123d 1251 . 2  |-  ( ph  ->  ( ( ps  /\  ch  /\  th )  -> 
( A. x ps 
/\  A. x ch  /\  A. x th ) ) )
5 19.26-3an 1413 . 2  |-  ( A. x ( ps  /\  ch  /\  th )  <->  ( A. x ps  /\  A. x ch  /\  A. x th ) )
64, 5syl6ibr 160 1  |-  ( ph  ->  ( ( ps  /\  ch  /\  th )  ->  A. x ( ps  /\  ch  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 920   A.wal 1283
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 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by: (None)
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