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Theorem 4casesdan 916
Description: Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
4casesdan.1  |-  ( (
ph  /\  ( ps  /\ 
ch ) )  ->  th )
4casesdan.2  |-  ( (
ph  /\  ( ps  /\ 
-.  ch ) )  ->  th )
4casesdan.3  |-  ( (
ph  /\  ( -.  ps  /\  ch ) )  ->  th )
4casesdan.4  |-  ( (
ph  /\  ( -.  ps  /\  -.  ch )
)  ->  th )
Assertion
Ref Expression
4casesdan  |-  ( ph  ->  th )

Proof of Theorem 4casesdan
StepHypRef Expression
1 4casesdan.1 . . 3  |-  ( (
ph  /\  ( ps  /\ 
ch ) )  ->  th )
21expcom 424 . 2  |-  ( ( ps  /\  ch )  ->  ( ph  ->  th )
)
3 4casesdan.2 . . 3  |-  ( (
ph  /\  ( ps  /\ 
-.  ch ) )  ->  th )
43expcom 424 . 2  |-  ( ( ps  /\  -.  ch )  ->  ( ph  ->  th ) )
5 4casesdan.3 . . 3  |-  ( (
ph  /\  ( -.  ps  /\  ch ) )  ->  th )
65expcom 424 . 2  |-  ( ( -.  ps  /\  ch )  ->  ( ph  ->  th ) )
7 4casesdan.4 . . 3  |-  ( (
ph  /\  ( -.  ps  /\  -.  ch )
)  ->  th )
87expcom 424 . 2  |-  ( ( -.  ps  /\  -.  ch )  ->  ( ph  ->  th ) )
92, 4, 6, 84cases 915 1  |-  ( ph  ->  th )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358
This theorem is referenced by:  unxpdomlem3  7069  cdleme41snaw  30665  dihord  31454  dihjat  31613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
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