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Theorem ifpdfxor 38162
Description: Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpdfxor  |-  ( (
ph  \/_  ps )  <-> if- (
ph ,  -.  ps ,  ps ) )

Proof of Theorem ifpdfxor
StepHypRef Expression
1 xor2 1369 . 2  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
2 ifpdfor 38144 . . 3  |-  ( (
ph  \/  ps )  <-> if- (
ph , T.  ,  ps ) )
3 ifpnot23 38135 . . . 4  |-  ( -. if- ( ph ,  ps , F.  )  <-> if- ( ph ,  -.  ps ,  -. F.  ) )
4 ifpdfan 38150 . . . 4  |-  ( (
ph  /\  ps )  <-> if- (
ph ,  ps , F.  ) )
53, 4xchnxbir 307 . . 3  |-  ( -.  ( ph  /\  ps ) 
<-> if- ( ph ,  -.  ps ,  -. F.  )
)
62, 5anbi12i 695 . 2  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  <->  (if- ( ph , T.  ,  ps )  /\ if- ( ph ,  -.  ps ,  -. F.  ) ) )
7 ifpan23 38155 . . 3  |-  ( (if- ( ph , T.  ,  ps )  /\ if- ( ph ,  -.  ps ,  -. F.  ) )  <-> if- ( ph , 
( T.  /\  -.  ps ) ,  ( ps 
/\  -. F.  )
) )
8 truan 1416 . . . 4  |-  ( ( T.  /\  -.  ps ) 
<->  -.  ps )
9 fal 1406 . . . . . 6  |-  -. F.
109biantru 503 . . . . 5  |-  ( ps  <->  ( ps  /\  -. F.  ) )
1110bicomi 202 . . . 4  |-  ( ( ps  /\  -. F.  ) 
<->  ps )
12 ifpbi23 38117 . . . 4  |-  ( ( ( ( T.  /\  -.  ps )  <->  -.  ps )  /\  ( ( ps  /\  -. F.  )  <->  ps )
)  ->  (if- ( ph ,  ( T.  /\  -.  ps ) ,  ( ps  /\  -. F.  ) )  <-> if- ( ph ,  -.  ps ,  ps )
) )
138, 11, 12mp2an 670 . . 3  |-  (if- (
ph ,  ( T. 
/\  -.  ps ) ,  ( ps  /\  -. F.  ) )  <-> if- ( ph ,  -.  ps ,  ps )
)
147, 13bitri 249 . 2  |-  ( (if- ( ph , T.  ,  ps )  /\ if- ( ph ,  -.  ps ,  -. F.  ) )  <-> if- ( ph ,  -.  ps ,  ps )
)
151, 6, 143bitri 271 1  |-  ( (
ph  \/_  ps )  <-> if- (
ph ,  -.  ps ,  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    \/_ wxo 1362  if-wif 1382   T. wtru 1400   F. wfal 1404
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 185  df-or 368  df-an 369  df-xor 1363  df-ifp 1383  df-tru 1402  df-fal 1405
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator