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Theorem ifpfal 1017
Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4041. This is essentially dedlemb 993. (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 1015 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2 ifptru 1016 . 2 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
31, 2syl5bb 270 1 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  if-wif 1005
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 195  df-or 383  df-an 384  df-ifp 1006
This theorem is referenced by:  ifpid  1018  elimh  1023  1wlkdlem4  40893  lfgriswlk  40896  2pthnloop  40936  eupth2lem3lem4  41398
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