MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.6 Structured version   Unicode version

Theorem pm5.6 880
Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.)
Assertion
Ref Expression
pm5.6  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  (
ph  ->  ( ps  \/  ch ) ) )

Proof of Theorem pm5.6
StepHypRef Expression
1 impexp 435 . 2  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  (
ph  ->  ( -.  ps  ->  ch ) ) )
2 df-or 361 . . 3  |-  ( ( ps  \/  ch )  <->  ( -.  ps  ->  ch ) )
32imbi2i 305 . 2  |-  ( (
ph  ->  ( ps  \/  ch ) )  <->  ( ph  ->  ( -.  ps  ->  ch ) ) )
41, 3bitr4i 245 1  |-  ( ( ( ph  /\  -.  ps )  ->  ch )  <->  (
ph  ->  ( ps  \/  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360
This theorem is referenced by:  ssundif  3713  brdom3  8411  grothprim  8714  eliccelico  24145  elicoelioo  24146  ballotlemfc0  24755  ballotlemfcc  24756  elicc3  26334
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
  Copyright terms: Public domain W3C validator