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Theorem ccased 911
Description: Deduction for combining cases. (Contributed by NM, 9-May-2004.)
Hypotheses
Ref Expression
ccased.1  |-  ( ph  ->  ( ( ps  /\  ch )  ->  et ) )
ccased.2  |-  ( ph  ->  ( ( th  /\  ch )  ->  et ) )
ccased.3  |-  ( ph  ->  ( ( ps  /\  ta )  ->  et ) )
ccased.4  |-  ( ph  ->  ( ( th  /\  ta )  ->  et ) )
Assertion
Ref Expression
ccased  |-  ( ph  ->  ( ( ( ps  \/  th )  /\  ( ch  \/  ta ) )  ->  et ) )

Proof of Theorem ccased
StepHypRef Expression
1 ccased.1 . . . 4  |-  ( ph  ->  ( ( ps  /\  ch )  ->  et ) )
21com12 30 . . 3  |-  ( ( ps  /\  ch )  ->  ( ph  ->  et ) )
3 ccased.2 . . . 4  |-  ( ph  ->  ( ( th  /\  ch )  ->  et ) )
43com12 30 . . 3  |-  ( ( th  /\  ch )  ->  ( ph  ->  et ) )
5 ccased.3 . . . 4  |-  ( ph  ->  ( ( ps  /\  ta )  ->  et ) )
65com12 30 . . 3  |-  ( ( ps  /\  ta )  ->  ( ph  ->  et ) )
7 ccased.4 . . . 4  |-  ( ph  ->  ( ( th  /\  ta )  ->  et ) )
87com12 30 . . 3  |-  ( ( th  /\  ta )  ->  ( ph  ->  et ) )
92, 4, 6, 8ccase 910 . 2  |-  ( ( ( ps  \/  th )  /\  ( ch  \/  ta ) )  ->  ( ph  ->  et ) )
109com12 30 1  |-  ( ph  ->  ( ( ( ps  \/  th )  /\  ( ch  \/  ta ) )  ->  et ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664
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-io 665
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  zmulcl  8801  gcdabs  11253  lcmabs  11332
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