MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xorass Structured version   Unicode version

Theorem xorass 1404
Description:  \/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.)
Assertion
Ref Expression
xorass  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  ( ph  \/_  ( ps  \/_  ch ) ) )

Proof of Theorem xorass
StepHypRef Expression
1 xor3 358 . . 3  |-  ( -.  ( ph  <->  ( ps  \/_ 
ch ) )  <->  ( ph  <->  -.  ( ps  \/_  ch ) ) )
2 biass 360 . . . 4  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
3 xnor 1402 . . . . 5  |-  ( (
ph 
<->  ps )  <->  -.  ( ph  \/_  ps ) )
43bibi1i 315 . . . 4  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( -.  ( ph  \/_  ps )  <->  ch )
)
5 xnor 1402 . . . . 5  |-  ( ( ps  <->  ch )  <->  -.  ( ps  \/_  ch ) )
65bibi2i 314 . . . 4  |-  ( (
ph 
<->  ( ps  <->  ch )
)  <->  ( ph  <->  -.  ( ps  \/_  ch ) ) )
72, 4, 63bitr3i 278 . . 3  |-  ( ( -.  ( ph  \/_  ps ) 
<->  ch )  <->  ( ph  <->  -.  ( ps  \/_  ch ) ) )
8 nbbn 359 . . 3  |-  ( ( -.  ( ph  \/_  ps ) 
<->  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
91, 7, 83bitr2ri 277 . 2  |-  ( -.  ( ( ph  \/_  ps ) 
<->  ch )  <->  -.  ( ph 
<->  ( ps  \/_  ch ) ) )
10 df-xor 1401 . 2  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  -.  (
( ph  \/_  ps )  <->  ch ) )
11 df-xor 1401 . 2  |-  ( (
ph  \/_  ( ps  \/_ 
ch ) )  <->  -.  ( ph 
<->  ( ps  \/_  ch ) ) )
129, 10, 113bitr4i 280 1  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  ( ph  \/_  ( ps  \/_  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/_ wxo 1400
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-xor 1401
This theorem is referenced by:  hadass  1495
  Copyright terms: Public domain W3C validator