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Theorem xoranor 1284
Description: One way of defining exclusive or. Equivalent to df-xor 1283. (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 1283 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
2 ax-ia3 105 . . . . . . 7  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
32con3d 571 . . . . . 6  |-  ( ph  ->  ( -.  ( ph  /\ 
ps )  ->  -.  ps ) )
4 olc 642 . . . . . 6  |-  ( -. 
ps  ->  ( -.  ph  \/  -.  ps ) )
53, 4syl6 33 . . . . 5  |-  ( ph  ->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) )
6 pm3.21 255 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
76con3d 571 . . . . . 6  |-  ( ps 
->  ( -.  ( ph  /\ 
ps )  ->  -.  ph ) )
8 orc 643 . . . . . 6  |-  ( -. 
ph  ->  ( -.  ph  \/  -.  ps ) )
97, 8syl6 33 . . . . 5  |-  ( ps 
->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) )
105, 9jaoi 646 . . . 4  |-  ( (
ph  \/  ps )  ->  ( -.  ( ph  /\ 
ps )  ->  ( -.  ph  \/  -.  ps ) ) )
1110imdistani 427 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
121, 11sylbi 118 . 2  |-  ( (
ph  \/_  ps )  ->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
13 pm3.14 680 . . . 4  |-  ( ( -.  ph  \/  -.  ps )  ->  -.  ( ph  /\  ps ) )
1413anim2i 328 . . 3  |-  ( ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) )  ->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
) )
1514, 1sylibr 141 . 2  |-  ( ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) )  ->  ( ph  \/_  ps ) )
1612, 15impbii 121 1  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ 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:  excxor  1285  xoror  1286
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