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Theorem pm5.17 896
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 203 . 2  |-  ( (
ph 
<->  -.  ps )  <->  ( -.  ps 
<-> 
ph ) )
2 dfbi2 632 . 2  |-  ( ( -.  ps  <->  ph )  <->  ( ( -.  ps  ->  ph )  /\  ( ph  ->  -.  ps )
) )
3 orcom 388 . . . 4  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
4 df-or 371 . . . 4  |-  ( ( ps  \/  ph )  <->  ( -.  ps  ->  ph )
)
53, 4bitr2i 253 . . 3  |-  ( ( -.  ps  ->  ph )  <->  (
ph  \/  ps )
)
6 imnan 423 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
75, 6anbi12i 701 . 2  |-  ( ( ( -.  ps  ->  ph )  /\  ( ph  ->  -.  ps ) )  <-> 
( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
81, 2, 73bitrri 275 1  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  <->  ( ph  <->  -. 
ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370
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 188  df-or 371  df-an 372
This theorem is referenced by:  nbi2  900  odd2np1  14353  ordtconlem1  28726  sgnneg  29407
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