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Theorem pm5.18 358
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 pm5.501 343 . . . 4  |-  ( ph  ->  ( -.  ps  <->  ( ph  <->  -. 
ps ) ) )
21con1bid 332 . . 3  |-  ( ph  ->  ( -.  ( ph  <->  -. 
ps )  <->  ps )
)
3 pm5.501 343 . . 3  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
42, 3bitr2d 258 . 2  |-  ( ph  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) )
5 nbn2 347 . . . 4  |-  ( -. 
ph  ->  ( -.  -.  ps 
<->  ( ph  <->  -.  ps )
) )
65con1bid 332 . . 3  |-  ( -. 
ph  ->  ( -.  ( ph 
<->  -.  ps )  <->  -.  ps )
)
7 nbn2 347 . . 3  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
86, 7bitr2d 258 . 2  |-  ( -. 
ph  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -. 
ps ) ) )
94, 8pm2.61i 168 1  |-  ( (
ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188
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 189
This theorem is referenced by:  xor3  359  pm5.19  362  pm5.16  901  dfbi3  904  xorassOLD  1410  xorneg2  1416
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