ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xorcom Unicode version

Theorem xorcom 1295
Description:  \/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
Assertion
Ref Expression
xorcom  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )

Proof of Theorem xorcom
StepHypRef Expression
1 orcom 657 . . 3  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
2 ancom 257 . . . 4  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
32notbii 604 . . 3  |-  ( -.  ( ph  /\  ps ) 
<->  -.  ( ps  /\  ph ) )
41, 3anbi12i 441 . 2  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  <->  ( ( ps  \/  ph )  /\  -.  ( ps  /\  ph ) ) )
5 df-xor 1283 . 2  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
6 df-xor 1283 . 2  |-  ( ( ps  \/_  ph )  <->  ( ( ps  \/  ph )  /\  -.  ( ps  /\  ph ) ) )
74, 5, 63bitr4i 205 1  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    <-> wb 102    \/ wo 639    \/_ wxo 1282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-xor 1283
This theorem is referenced by:  rpnegap  8713
  Copyright terms: Public domain W3C validator