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Theorem ifpfal 1432
Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 3924. This is essentially dedlemb 963. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
Assertion
Ref Expression
ifpfal  |-  ( -. 
ph  ->  (if- ( ph ,  ps ,  ch )  <->  ch ) )

Proof of Theorem ifpfal
StepHypRef Expression
1 ifpn 1430 . 2  |-  (if- (
ph ,  ps ,  ch )  <-> if- ( -.  ph ,  ch ,  ps )
)
2 ifptru 1431 . 2  |-  ( -. 
ph  ->  (if- ( -. 
ph ,  ch ,  ps )  <->  ch ) )
31, 2syl5bb 260 1  |-  ( -. 
ph  ->  (if- ( ph ,  ps ,  ch )  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187  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:  ifpid  1433  bj-elimhyp  30938
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