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Theorem dfifp6 1426
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
Assertion
Ref Expression
dfifp6  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  -.  ( ch  ->  ph )
) )

Proof of Theorem dfifp6
StepHypRef Expression
1 df-ifp 1421 . 2  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  ch )
) )
2 ancom 451 . . . 4  |-  ( ( -.  ph  /\  ch )  <->  ( ch  /\  -.  ph ) )
3 annim 426 . . . 4  |-  ( ( ch  /\  -.  ph ) 
<->  -.  ( ch  ->  ph ) )
42, 3bitri 252 . . 3  |-  ( ( -.  ph  /\  ch )  <->  -.  ( ch  ->  ph )
)
54orbi2i 521 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( ph  /\  ps )  \/ 
-.  ( ch  ->  ph ) ) )
61, 5bitri 252 1  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  -.  ( ch  ->  ph )
) )
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:  dfifp7  1427  ifpdfan2  35805
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