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Theorem dfifp3 1423
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
dfifp3  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch ) ) )

Proof of Theorem dfifp3
StepHypRef Expression
1 dfifp2 1422 . 2  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch )
) )
2 pm4.64 373 . . 3  |-  ( ( -.  ph  ->  ch )  <->  (
ph  \/  ch )
)
32anbi2i 698 . 2  |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ch ) )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch )
) )
41, 3bitri 252 1  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370  if-wif 1420
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
This theorem is referenced by:  dfifp4  1424  ifptru  1431
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