Theorem ifpfal 1073

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4530. This is essentially dedlemb 1043. (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

Step	Hyp	Ref	Expression
1		ifpn 1070	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1072	. 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
3	1, 2	bitrid 283	1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  if-wif 1059

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 206  df-an 396  df-or 845  df-ifp 1060

This theorem is referenced by:  ifpid  1074  elimh  1080  axprlem3  5414  axprlem5  5416  wlkdlem4  29437  lfgriswlk  29440  2pthnloop  29483  eupth2lem3lem4  29979  satfv1lem  34871  wl-3xorfal  36854  sn-axprlem3  41569