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

Theorem ccased 938
Description: Deduction for combining cases. (Contributed by NM, 9-May-2004.)
Hypotheses
Ref Expression
ccased.1  |-  ( ph  ->  ( ( ps  /\  ch )  ->  et ) )
ccased.2  |-  ( ph  ->  ( ( th  /\  ch )  ->  et ) )
ccased.3  |-  ( ph  ->  ( ( ps  /\  ta )  ->  et ) )
ccased.4  |-  ( ph  ->  ( ( th  /\  ta )  ->  et ) )
Assertion
Ref Expression
ccased  |-  ( ph  ->  ( ( ( ps  \/  th )  /\  ( ch  \/  ta ) )  ->  et ) )

Proof of Theorem ccased
StepHypRef Expression
1 ccased.1 . . . 4  |-  ( ph  ->  ( ( ps  /\  ch )  ->  et ) )
21com12 31 . . 3  |-  ( ( ps  /\  ch )  ->  ( ph  ->  et ) )
3 ccased.2 . . . 4  |-  ( ph  ->  ( ( th  /\  ch )  ->  et ) )
43com12 31 . . 3  |-  ( ( th  /\  ch )  ->  ( ph  ->  et ) )
5 ccased.3 . . . 4  |-  ( ph  ->  ( ( ps  /\  ta )  ->  et ) )
65com12 31 . . 3  |-  ( ( ps  /\  ta )  ->  ( ph  ->  et ) )
7 ccased.4 . . . 4  |-  ( ph  ->  ( ( th  /\  ta )  ->  et ) )
87com12 31 . . 3  |-  ( ( th  /\  ta )  ->  ( ph  ->  et ) )
92, 4, 6, 8ccase 937 . 2  |-  ( ( ( ps  \/  th )  /\  ( ch  \/  ta ) )  ->  ( ph  ->  et ) )
109com12 31 1  |-  ( ph  ->  ( ( ( ps  \/  th )  /\  ( ch  \/  ta ) )  ->  et ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369
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 185  df-or 370  df-an 371
This theorem is referenced by:  fpwwe2lem13  8828  mulge0  9876  zmulcl  10712  gcdabs  13736  pospo  15162  mulgass  15676  indistopon  18624  lgsdir2lem5  22685  outsideofeq  28180  smprngopr  28875  monotoddzzfi  29306  acongtr  29344  cdlemg33  34378
  Copyright terms: Public domain W3C validator