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Theorem ifptru 1431
Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 3921. This is essentially dedlema 962. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
Assertion
Ref Expression
ifptru  |-  ( ph  ->  (if- ( ph ,  ps ,  ch )  <->  ps ) )

Proof of Theorem ifptru
StepHypRef Expression
1 biimt 336 . 2  |-  ( ph  ->  ( ps  <->  ( ph  ->  ps ) ) )
2 orc 386 . . . 4  |-  ( ph  ->  ( ph  \/  ch ) )
32biantrud 509 . . 3  |-  ( ph  ->  ( ( ph  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch ) ) ) )
4 dfifp3 1423 . . 3  |-  (if- (
ph ,  ps ,  ch )  <->  ( ( ph  ->  ps )  /\  ( ph  \/  ch ) ) )
53, 4syl6bbr 266 . 2  |-  ( ph  ->  ( ( ph  ->  ps )  <-> if- ( ph ,  ps ,  ch ) ) )
61, 5bitr2d 257 1  |-  ( ph  ->  (if- ( ph ,  ps ,  ch )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> 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:  ifpfal  1432  ifpid  1433  bj-elimhyp  30938  bj-dedthm  30939
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