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Theorem pm5.17 858
Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
Assertion
Ref Expression
pm5.17  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  <->  ( ph  <->  -. 
ps ) )

Proof of Theorem pm5.17
StepHypRef Expression
1 bicom 191 . 2  |-  ( (
ph 
<->  -.  ps )  <->  ( -.  ps 
<-> 
ph ) )
2 dfbi2 609 . 2  |-  ( ( -.  ps  <->  ph )  <->  ( ( -.  ps  ->  ph )  /\  ( ph  ->  -.  ps )
) )
3 orcom 376 . . . 4  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
4 df-or 359 . . . 4  |-  ( ( ps  \/  ph )  <->  ( -.  ps  ->  ph )
)
53, 4bitr2i 241 . . 3  |-  ( ( -.  ps  ->  ph )  <->  (
ph  \/  ps )
)
6 imnan 411 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
75, 6anbi12i 678 . 2  |-  ( ( ( -.  ps  ->  ph )  /\  ( ph  ->  -.  ps ) )  <-> 
( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
81, 2, 73bitrri 263 1  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  <->  ( ph  <->  -. 
ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  nbi2  862  odd2np1  12587
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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