ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm4.72 Unicode version

Theorem pm4.72 736
Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)
Assertion
Ref Expression
pm4.72  |-  ( (
ph  ->  ps )  <->  ( ps  <->  (
ph  \/  ps )
) )

Proof of Theorem pm4.72
StepHypRef Expression
1 olc 632 . . 3  |-  ( ps 
->  ( ph  \/  ps ) )
2 pm2.621 666 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ph  \/  ps )  ->  ps )
)
31, 2impbid2 131 . 2  |-  ( (
ph  ->  ps )  -> 
( ps  <->  ( ph  \/  ps ) ) )
4 orc 633 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
5 bi2 121 . . 3  |-  ( ( ps  <->  ( ph  \/  ps ) )  ->  (
( ph  \/  ps )  ->  ps ) )
64, 5syl5 28 . 2  |-  ( ( ps  <->  ( ph  \/  ps ) )  ->  ( ph  ->  ps ) )
73, 6impbii 117 1  |-  ( (
ph  ->  ps )  <->  ( ps  <->  (
ph  \/  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    \/ wo 629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  bigolden  862  ssequn1  3113
  Copyright terms: Public domain W3C validator