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Theorem ifpfal 1018
 Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4045. This is essentially dedlemb 994. (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 1016 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2 ifptru 1017 . 2 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
31, 2syl5bb 271 1 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  if-wif 1006 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 196  df-or 384  df-an 385  df-ifp 1007 This theorem is referenced by:  ifpid  1019  elimh  1024  1wlkdlem4  40894  lfgriswlk  40897  2pthnloop  40937  eupth2lem3lem4  41399
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