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Theorem 3jaao 1296
Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Hypotheses
Ref Expression
3jaao.1  |-  ( ph  ->  ( ps  ->  ch ) )
3jaao.2  |-  ( th 
->  ( ta  ->  ch ) )
3jaao.3  |-  ( et 
->  ( ze  ->  ch ) )
Assertion
Ref Expression
3jaao  |-  ( (
ph  /\  th  /\  et )  ->  ( ( ps  \/  ta  \/  ze )  ->  ch ) )

Proof of Theorem 3jaao
StepHypRef Expression
1 3jaao.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
213ad2ant1 1017 . 2  |-  ( (
ph  /\  th  /\  et )  ->  ( ps  ->  ch ) )
3 3jaao.2 . . 3  |-  ( th 
->  ( ta  ->  ch ) )
433ad2ant2 1018 . 2  |-  ( (
ph  /\  th  /\  et )  ->  ( ta  ->  ch ) )
5 3jaao.3 . . 3  |-  ( et 
->  ( ze  ->  ch ) )
653ad2ant3 1019 . 2  |-  ( (
ph  /\  th  /\  et )  ->  ( ze  ->  ch ) )
72, 4, 63jaod 1292 1  |-  ( (
ph  /\  th  /\  et )  ->  ( ( ps  \/  ta  \/  ze )  ->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 972    /\ w3a 973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975
This theorem is referenced by:  lpni  24857  3ornot23  32357
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