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Theorem xordc1 1300
Description: Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
Assertion
Ref Expression
xordc1  |-  ( (
ph  \/_  ps )  -> DECID  ph )

Proof of Theorem xordc1
StepHypRef Expression
1 andir 743 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  <->  ( ( ph  /\  -.  ( ph  /\ 
ps ) )  \/  ( ps  /\  -.  ( ph  /\  ps )
) ) )
2 simpl 106 . . . 4  |-  ( (
ph  /\  -.  ( ph  /\  ps ) )  ->  ph )
3 imnan 634 . . . . . 6  |-  ( ( ps  ->  -.  ph )  <->  -.  ( ps  /\  ph ) )
4 ancom 257 . . . . . 6  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
53, 4xchbinxr 618 . . . . 5  |-  ( ( ps  ->  -.  ph )  <->  -.  ( ph  /\  ps ) )
6 pm3.35 333 . . . . 5  |-  ( ( ps  /\  ( ps 
->  -.  ph ) )  ->  -.  ph )
75, 6sylan2br 276 . . . 4  |-  ( ( ps  /\  -.  ( ph  /\  ps ) )  ->  -.  ph )
82, 7orim12i 686 . . 3  |-  ( ( ( ph  /\  -.  ( ph  /\  ps )
)  \/  ( ps 
/\  -.  ( ph  /\ 
ps ) ) )  ->  ( ph  \/  -.  ph ) )
91, 8sylbi 118 . 2  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( ph  \/  -.  ph ) )
10 df-xor 1283 . 2  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
11 df-dc 754 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
129, 10, 113imtr4i 194 1  |-  ( (
ph  \/_  ps )  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639  DECID wdc 753    \/_ 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-dc 754  df-xor 1283
This theorem is referenced by: (None)
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