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

Theorem xorcom 1403
Description:  \/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
xorcom  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )

Proof of Theorem xorcom
StepHypRef Expression
1 bicom 203 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )
21notbii 297 . 2  |-  ( -.  ( ph  <->  ps )  <->  -.  ( ps  <->  ph ) )
3 df-xor 1401 . 2  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
4 df-xor 1401 . 2  |-  ( ( ps  \/_  ph )  <->  -.  ( ps 
<-> 
ph ) )
52, 3, 43bitr4i 280 1  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )
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:  xorneg1  1412  xorneg2OLD  1414  falxortru  1488  hadcoma  1497  hadcomb  1498  cadcoma  1510
  Copyright terms: Public domain W3C validator