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Theorem 3jaao 1214
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 936 . 2  |-  ( (
ph  /\  th  /\  et )  ->  ( ps  ->  ch ) )
3 3jaao.2 . . 3  |-  ( th 
->  ( ta  ->  ch ) )
433ad2ant2 937 . 2  |-  ( (
ph  /\  th  /\  et )  ->  ( ta  ->  ch ) )
5 3jaao.3 . . 3  |-  ( et 
->  ( ze  ->  ch ) )
653ad2ant3 938 . 2  |-  ( (
ph  /\  th  /\  et )  ->  ( ze  ->  ch ) )
72, 4, 63jaod 1210 1  |-  ( (
ph  /\  th  /\  et )  ->  ( ( ps  \/  ta  \/  ze )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 895    /\ w3a 896
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-io 640
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898
This theorem is referenced by: (None)
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