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Theorem xorass 1299
Description:  \/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
xorass  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  ( ph  \/_  ( ps  \/_  ch ) ) )

Proof of Theorem xorass
StepHypRef Expression
1 biass 348 . . . . . 6  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
21notbii 287 . . . . 5  |-  ( -.  ( ( ph  <->  ps )  <->  ch )  <->  -.  ( ph  <->  ( ps  <->  ch ) ) )
3 nbbn 347 . . . . 5  |-  ( ( -.  ( ph  <->  ps )  <->  ch )  <->  -.  ( ( ph 
<->  ps )  <->  ch )
)
4 pm5.18 345 . . . . . 6  |-  ( (
ph 
<->  ( ps  <->  ch )
)  <->  -.  ( ph  <->  -.  ( ps  <->  ch )
) )
54con2bii 322 . . . . 5  |-  ( (
ph 
<->  -.  ( ps  <->  ch )
)  <->  -.  ( ph  <->  ( ps  <->  ch ) ) )
62, 3, 53bitr4i 268 . . . 4  |-  ( ( -.  ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  -.  ( ps 
<->  ch ) ) )
7 df-xor 1296 . . . . 5  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
87bibi1i 305 . . . 4  |-  ( ( ( ph  \/_  ps ) 
<->  ch )  <->  ( -.  ( ph  <->  ps )  <->  ch )
)
9 df-xor 1296 . . . . 5  |-  ( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch )
)
109bibi2i 304 . . . 4  |-  ( (
ph 
<->  ( ps  \/_  ch ) )  <->  ( ph  <->  -.  ( ps  <->  ch )
) )
116, 8, 103bitr4i 268 . . 3  |-  ( ( ( ph  \/_  ps ) 
<->  ch )  <->  ( ph  <->  ( ps  \/_  ch )
) )
1211notbii 287 . 2  |-  ( -.  ( ( ph  \/_  ps ) 
<->  ch )  <->  -.  ( ph 
<->  ( ps  \/_  ch ) ) )
13 df-xor 1296 . 2  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
14 df-xor 1296 . 2  |-  ( (
ph  \/_  ( ps  \/_ 
ch ) )  <->  -.  ( ph 
<->  ( ps  \/_  ch ) ) )
1512, 13, 143bitr4i 268 1  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  ( ph  \/_  ( ps  \/_  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/_ wxo 1295
This theorem is referenced by:  hadass  1377
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-xor 1296
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