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Theorem ifpim1 35811
Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpim1  |-  ( (
ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )

Proof of Theorem ifpim1
StepHypRef Expression
1 tru 1441 . . . 4  |- T.
21olci 392 . . 3  |-  ( -. 
-.  ph  \/ T.  )
32biantrur 508 . 2  |-  ( ( -.  ph  \/  ps ) 
<->  ( ( -.  -.  ph  \/ T.  )  /\  ( -.  ph  \/  ps ) ) )
4 imor 413 . 2  |-  ( (
ph  ->  ps )  <->  ( -.  ph  \/  ps ) )
5 dfifp4 1424 . 2  |-  (if- ( -.  ph , T.  ,  ps )  <->  ( ( -.  -.  ph  \/ T.  )  /\  ( -.  ph  \/  ps ) ) )
63, 4, 53bitr4i 280 1  |-  ( (
ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ 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:  ifpdfbi  35816
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