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Theorem pm5.75 904
Description: Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)
Assertion
Ref Expression
pm5.75  |-  ( ( ( ch  ->  -.  ps )  /\  ( ph 
<->  ( ps  \/  ch ) ) )  -> 
( ( ph  /\  -.  ps )  <->  ch )
)

Proof of Theorem pm5.75
StepHypRef Expression
1 anbi1 688 . . 3  |-  ( (
ph 
<->  ( ps  \/  ch ) )  ->  (
( ph  /\  -.  ps ) 
<->  ( ( ps  \/  ch )  /\  -.  ps ) ) )
2 orcom 377 . . . . 5  |-  ( ( ps  \/  ch )  <->  ( ch  \/  ps )
)
32anbi1i 677 . . . 4  |-  ( ( ( ps  \/  ch )  /\  -.  ps )  <->  ( ( ch  \/  ps )  /\  -.  ps )
)
4 pm5.61 694 . . . 4  |-  ( ( ( ch  \/  ps )  /\  -.  ps )  <->  ( ch  /\  -.  ps ) )
53, 4bitri 241 . . 3  |-  ( ( ( ps  \/  ch )  /\  -.  ps )  <->  ( ch  /\  -.  ps ) )
61, 5syl6bb 253 . 2  |-  ( (
ph 
<->  ( ps  \/  ch ) )  ->  (
( ph  /\  -.  ps ) 
<->  ( ch  /\  -.  ps ) ) )
7 pm4.71 612 . . . 4  |-  ( ( ch  ->  -.  ps )  <->  ( ch  <->  ( ch  /\  -.  ps ) ) )
87biimpi 187 . . 3  |-  ( ( ch  ->  -.  ps )  ->  ( ch  <->  ( ch  /\ 
-.  ps ) ) )
98bicomd 193 . 2  |-  ( ( ch  ->  -.  ps )  ->  ( ( ch  /\  -.  ps )  <->  ch )
)
106, 9sylan9bbr 682 1  |-  ( ( ( ch  ->  -.  ps )  /\  ( ph 
<->  ( ps  \/  ch ) ) )  -> 
( ( ph  /\  -.  ps )  <->  ch )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359
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 178  df-or 360  df-an 361
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