Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpdfnan Structured version   Unicode version

Theorem ifpdfnan 36050
Description: Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpdfnan  |-  ( (
ph  -/\  ps )  <-> if- ( ph ,  -.  ps , T.  ) )

Proof of Theorem ifpdfnan
StepHypRef Expression
1 df-nan 1380 . 2  |-  ( (
ph  -/\  ps )  <->  -.  ( ph  /\  ps ) )
2 ifpdfan 36029 . . 3  |-  ( (
ph  /\  ps )  <-> if- (
ph ,  ps , F.  ) )
32notbii 297 . 2  |-  ( -.  ( ph  /\  ps ) 
<->  -. if- ( ph ,  ps , F.  )
)
4 ifpnot23 36042 . . 3  |-  ( -. if- ( ph ,  ps , F.  )  <-> if- ( ph ,  -.  ps ,  -. F.  ) )
5 notfal 1474 . . . 4  |-  ( -. F.  <-> T.  )
6 ifpbi3 36031 . . . 4  |-  ( ( -. F.  <-> T.  )  ->  (if- ( ph ,  -.  ps ,  -. F.  ) 
<-> if- ( ph ,  -.  ps , T.  )
) )
75, 6ax-mp 5 . . 3  |-  (if- (
ph ,  -.  ps ,  -. F.  )  <-> if- ( ph ,  -.  ps , T.  ) )
84, 7bitri 252 . 2  |-  ( -. if- ( ph ,  ps , F.  )  <-> if- ( ph ,  -.  ps , T.  ) )
91, 3, 83bitri 274 1  |-  ( (
ph  -/\  ps )  <-> if- ( ph ,  -.  ps , T.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    -/\ wnan 1379  if-wif 1420   T. wtru 1438   F. wfal 1442
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 188  df-or 371  df-an 372  df-nan 1380  df-ifp 1421  df-tru 1440  df-fal 1443
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator