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Theorem ifpdfbi 36087
Description: Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpdfbi  |-  ( (
ph 
<->  ps )  <-> if- ( ph ,  ps ,  -.  ps )
)

Proof of Theorem ifpdfbi
StepHypRef Expression
1 dfbi2 632 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
2 ifpim1 36082 . . . . 5  |-  ( (
ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )
3 ifpn 1430 . . . . 5  |-  (if- (
ph ,  ps , T.  )  <-> if- ( -.  ph , T.  ,  ps )
)
42, 3bitr4i 255 . . . 4  |-  ( (
ph  ->  ps )  <-> if- ( ph ,  ps , T.  )
)
5 ifpim2 36085 . . . 4  |-  ( ( ps  ->  ph )  <-> if- ( ph , T.  ,  -.  ps )
)
64, 5anbi12i 701 . . 3  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  (if- ( ph ,  ps , T.  )  /\ if- ( ph , T.  ,  -.  ps )
) )
7 ifpan23 36073 . . . 4  |-  ( (if- ( ph ,  ps , T.  )  /\ if- ( ph , T.  ,  -.  ps ) )  <-> if- ( ph , 
( ps  /\ T.  ) ,  ( T.  /\  -.  ps ) ) )
8 ancom 451 . . . . . 6  |-  ( ( ps  /\ T.  )  <-> 
( T.  /\  ps ) )
9 truan 1454 . . . . . 6  |-  ( ( T.  /\  ps )  <->  ps )
108, 9bitri 252 . . . . 5  |-  ( ( ps  /\ T.  )  <->  ps )
11 truan 1454 . . . . 5  |-  ( ( T.  /\  -.  ps ) 
<->  -.  ps )
12 ifpbi23 36086 . . . . 5  |-  ( ( ( ( ps  /\ T.  )  <->  ps )  /\  (
( T.  /\  -.  ps )  <->  -.  ps )
)  ->  (if- ( ph ,  ( ps  /\ T.  ) ,  ( T.  /\  -.  ps ) )  <-> if- ( ph ,  ps ,  -.  ps )
) )
1310, 11, 12mp2an 676 . . . 4  |-  (if- (
ph ,  ( ps 
/\ T.  ) ,  ( T.  /\  -.  ps ) )  <-> if- ( ph ,  ps ,  -.  ps )
)
147, 13bitri 252 . . 3  |-  ( (if- ( ph ,  ps , T.  )  /\ if- ( ph , T.  ,  -.  ps ) )  <-> if- ( ph ,  ps ,  -.  ps )
)
156, 14bitri 252 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <-> if- ( ph ,  ps ,  -.  ps )
)
161, 15bitri 252 1  |-  ( (
ph 
<->  ps )  <-> if- ( ph ,  ps ,  -.  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370  if-wif 1420   T. wtru 1438
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-ifp 1421  df-tru 1440
This theorem is referenced by:  ifpbiidcor  36088  ifpbicor  36089
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