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Theorem pm4.72 877
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 382 . . 3  |-  ( ps 
->  ( ph  \/  ps ) )
2 pm2.621 406 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ph  \/  ps )  ->  ps )
)
31, 2impbid2 204 . 2  |-  ( (
ph  ->  ps )  -> 
( ps  <->  ( ph  \/  ps ) ) )
4 orc 383 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
5 bi2 198 . . 3  |-  ( ( ps  <->  ( ph  \/  ps ) )  ->  (
( ph  \/  ps )  ->  ps ) )
64, 5syl5 30 . 2  |-  ( ( ps  <->  ( ph  \/  ps ) )  ->  ( ph  ->  ps ) )
73, 6impbii 188 1  |-  ( (
ph  ->  ps )  <->  ( ps  <->  (
ph  \/  ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366
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 368
This theorem is referenced by:  bigolden  936  cadan  1474  ssequn1  3612  ssunsn2  4130  elpaddn0  32797
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