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Theorem pm5.18 345
Description: Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or." (Contributed by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)
Assertion
Ref Expression
pm5.18  |-  ( (
ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) )

Proof of Theorem pm5.18
StepHypRef Expression
1 pm5.501 330 . . . 4  |-  ( ph  ->  ( -.  ps  <->  ( ph  <->  -. 
ps ) ) )
21con1bid 320 . . 3  |-  ( ph  ->  ( -.  ( ph  <->  -. 
ps )  <->  ps )
)
3 pm5.501 330 . . 3  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
42, 3bitr2d 245 . 2  |-  ( ph  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) )
5 nbn2 334 . . . 4  |-  ( -. 
ph  ->  ( -.  -.  ps 
<->  ( ph  <->  -.  ps )
) )
65con1bid 320 . . 3  |-  ( -. 
ph  ->  ( -.  ( ph 
<->  -.  ps )  <->  -.  ps )
)
7 nbn2 334 . . 3  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
86, 7bitr2d 245 . 2  |-  ( -. 
ph  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -. 
ps ) ) )
94, 8pm2.61i 156 1  |-  ( (
ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176
This theorem is referenced by:  xor3  346  pm5.19  349  pm5.16  860  dfbi3  863  xorass  1299
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
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