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Theorem pm5.54 870
Description: Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.)
Assertion
Ref Expression
pm5.54  |-  ( ( ( ph  /\  ps ) 
<-> 
ph )  \/  (
( ph  /\  ps )  <->  ps ) )

Proof of Theorem pm5.54
StepHypRef Expression
1 iba 489 . . . . 5  |-  ( ps 
->  ( ph  <->  ( ph  /\ 
ps ) ) )
21bicomd 192 . . . 4  |-  ( ps 
->  ( ( ph  /\  ps )  <->  ph ) )
32adantl 452 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ph  /\  ps )  <->  ph ) )
43, 2pm5.21ni 341 . 2  |-  ( -.  ( ( ph  /\  ps )  <->  ph )  ->  (
( ph  /\  ps )  <->  ps ) )
54orri 365 1  |-  ( ( ( ph  /\  ps ) 
<-> 
ph )  \/  (
( ph  /\  ps )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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