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Theorem 3ecase 1286
Description: Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
Hypotheses
Ref Expression
3ecase.1  |-  ( -. 
ph  ->  th )
3ecase.2  |-  ( -. 
ps  ->  th )
3ecase.3  |-  ( -. 
ch  ->  th )
3ecase.4  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
Assertion
Ref Expression
3ecase  |-  th

Proof of Theorem 3ecase
StepHypRef Expression
1 3ecase.4 . . . 4  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
213exp 1150 . . 3  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
3 3ecase.1 . . . . 5  |-  ( -. 
ph  ->  th )
43a1d 22 . . . 4  |-  ( -. 
ph  ->  ( ch  ->  th ) )
54a1d 22 . . 3  |-  ( -. 
ph  ->  ( ps  ->  ( ch  ->  th )
) )
62, 5pm2.61i 156 . 2  |-  ( ps 
->  ( ch  ->  th )
)
7 3ecase.2 . 2  |-  ( -. 
ps  ->  th )
8 3ecase.3 . 2  |-  ( -. 
ch  ->  th )
96, 7, 8pm2.61nii 158 1  |-  th
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934
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  df-3an 936
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