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Theorem ifpfal 1076
Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4534. This is essentially dedlemb 1047. (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 1074 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2 ifptru 1075 . 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 1063
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 849  df-ifp 1064
This theorem is referenced by:  ifpid  1077  elimh  1083  axprlem3  5425  axpr  5427  axprlem3OLD  5428  axprlem5OLD  5430  wlkdlem4  29703  lfgriswlk  29706  2pthnloop  29751  eupth2lem3lem4  30250  satfv1lem  35367  wl-3xorfal  37473  sn-axprlem3  42256
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