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Theorem pm4.72 848
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 375 . . 3  |-  ( ps 
->  ( ph  \/  ps ) )
2 pm2.621 399 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ph  \/  ps )  ->  ps )
)
31, 2impbid2 197 . 2  |-  ( (
ph  ->  ps )  -> 
( ps  <->  ( ph  \/  ps ) ) )
4 orc 376 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
5 bi2 191 . . 3  |-  ( ( ps  <->  ( ph  \/  ps ) )  ->  (
( ph  \/  ps )  ->  ps ) )
64, 5syl5 30 . 2  |-  ( ( ps  <->  ( ph  \/  ps ) )  ->  ( ph  ->  ps ) )
73, 6impbii 182 1  |-  ( (
ph  ->  ps )  <->  ( ps  <->  (
ph  \/  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359
This theorem is referenced by:  bigolden  903  ssequn1  3348  ssunsn2  3776  elpaddn0  29258
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361
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