MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecase23d Structured version   Unicode version

Theorem ecase23d 1288
Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)
Hypotheses
Ref Expression
ecase23d.1  |-  ( ph  ->  -.  ch )
ecase23d.2  |-  ( ph  ->  -.  th )
ecase23d.3  |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
Assertion
Ref Expression
ecase23d  |-  ( ph  ->  ps )

Proof of Theorem ecase23d
StepHypRef Expression
1 ecase23d.1 . . 3  |-  ( ph  ->  -.  ch )
2 ecase23d.2 . . 3  |-  ( ph  ->  -.  th )
3 ioran 478 . . 3  |-  ( -.  ( ch  \/  th ) 
<->  ( -.  ch  /\  -.  th ) )
41, 2, 3sylanbrc 647 . 2  |-  ( ph  ->  -.  ( ch  \/  th ) )
5 ecase23d.3 . . . 4  |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
6 3orass 940 . . . 4  |-  ( ( ps  \/  ch  \/  th )  <->  ( ps  \/  ( ch  \/  th )
) )
75, 6sylib 190 . . 3  |-  ( ph  ->  ( ps  \/  ( ch  \/  th ) ) )
87ord 368 . 2  |-  ( ph  ->  ( -.  ps  ->  ( ch  \/  th )
) )
94, 8mt3d 120 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    \/ w3o 936
This theorem is referenced by:  tz7.7  4610  wfrlem10  25552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938
  Copyright terms: Public domain W3C validator