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

Theorem ecased 952
Description: Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)
Hypotheses
Ref Expression
ecased.1  |-  ( ph  ->  ( -.  ps  ->  th ) )
ecased.2  |-  ( ph  ->  ( -.  ch  ->  th ) )
ecased.3  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
Assertion
Ref Expression
ecased  |-  ( ph  ->  th )

Proof of Theorem ecased
StepHypRef Expression
1 ecased.1 . 2  |-  ( ph  ->  ( -.  ps  ->  th ) )
2 ecased.2 . 2  |-  ( ph  ->  ( -.  ch  ->  th ) )
3 pm3.11 501 . . 3  |-  ( -.  ( -.  ps  \/  -.  ch )  ->  ( ps  /\  ch ) )
4 ecased.3 . . 3  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
53, 4syl5 33 . 2  |-  ( ph  ->  ( -.  ( -. 
ps  \/  -.  ch )  ->  th ) )
61, 2, 5ecase3d 951 1  |-  ( ph  ->  th )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370
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 188  df-or 371  df-an 372
This theorem is referenced by:  ecase3ad  953  itgsplitioo  22782  rolle  22929  dalaw  33370
  Copyright terms: Public domain W3C validator