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Theorem ifpfal 1075
Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4539. This is essentially dedlemb 1046. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
Assertion
Ref Expression
ifpfal 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Proof of Theorem ifpfal
StepHypRef Expression
1 ifpn 1073 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2 ifptru 1074 . 2 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
31, 2bitrid 283 1 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  if-wif 1062
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 207  df-an 396  df-or 848  df-ifp 1063
This theorem is referenced by:  ifpid  1076  elimh  1082  axprlem3  5430  axpr  5432  axprlem3OLD  5433  axprlem5OLD  5435  wlkdlem4  29717  lfgriswlk  29720  2pthnloop  29763  eupth2lem3lem4  30259  satfv1lem  35346  wl-3xorfal  37454  sn-axprlem3  42234
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