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Theorem ccase 908
Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
Hypotheses
Ref Expression
ccase.1  |-  ( (
ph  /\  ps )  ->  ta )
ccase.2  |-  ( ( ch  /\  ps )  ->  ta )
ccase.3  |-  ( (
ph  /\  th )  ->  ta )
ccase.4  |-  ( ( ch  /\  th )  ->  ta )
Assertion
Ref Expression
ccase  |-  ( ( ( ph  \/  ch )  /\  ( ps  \/  th ) )  ->  ta )

Proof of Theorem ccase
StepHypRef Expression
1 ccase.1 . . 3  |-  ( (
ph  /\  ps )  ->  ta )
2 ccase.2 . . 3  |-  ( ( ch  /\  ps )  ->  ta )
31, 2jaoian 742 . 2  |-  ( ( ( ph  \/  ch )  /\  ps )  ->  ta )
4 ccase.3 . . 3  |-  ( (
ph  /\  th )  ->  ta )
5 ccase.4 . . 3  |-  ( ( ch  /\  th )  ->  ta )
64, 5jaoian 742 . 2  |-  ( ( ( ph  \/  ch )  /\  th )  ->  ta )
73, 6jaodan 744 1  |-  ( ( ( ph  \/  ch )  /\  ( ps  \/  th ) )  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662
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 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  ccased  909  ccase2  910  undif3ss  3249  nn0gcdsq  11084
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