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

Theorem xoranor 1356
Description: One way of defining exclusive or. Equivalent to df-xor 1355. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
Assertion
Ref Expression
xoranor  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )

Proof of Theorem xoranor
StepHypRef Expression
1 df-xor 1355 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
2 ax-ia3 107 . . . . . . 7  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
32con3d 621 . . . . . 6  |-  ( ph  ->  ( -.  ( ph  /\ 
ps )  ->  -.  ps ) )
4 olc 701 . . . . . 6  |-  ( -. 
ps  ->  ( -.  ph  \/  -.  ps ) )
53, 4syl6 33 . . . . 5  |-  ( ph  ->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) )
6 pm3.21 262 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
76con3d 621 . . . . . 6  |-  ( ps 
->  ( -.  ( ph  /\ 
ps )  ->  -.  ph ) )
8 orc 702 . . . . . 6  |-  ( -. 
ph  ->  ( -.  ph  \/  -.  ps ) )
97, 8syl6 33 . . . . 5  |-  ( ps 
->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) )
105, 9jaoi 706 . . . 4  |-  ( (
ph  \/  ps )  ->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) )
1110imdistani 442 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
121, 11sylbi 120 . 2  |-  ( (
ph  \/_  ps )  ->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
13 pm3.14 743 . . . 4  |-  ( ( -.  ph  \/  -.  ps )  ->  -.  ( ph  /\  ps ) )
1413anim2i 340 . . 3  |-  ( ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) )  ->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
) )
1514, 1sylibr 133 . 2  |-  ( ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) )  ->  ( ph  \/_  ps ) )
1612, 15impbii 125 1  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    \/_ wxo 1354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-xor 1355
This theorem is referenced by:  excxor  1357  xoror  1358
  Copyright terms: Public domain W3C validator