Theorem ifpfal 1088

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4491. This is essentially dedlemb 1058. (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 1086	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1087	. 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
3	1, 2	bitrid 285	1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  if-wif 1074

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 209  df-an 400  df-or 859  df-ifp 1075

This theorem is referenced by:  ifpid  1089  elimh  1095  axprlem3  5384  axpr  5386  axprlem3OLD  5388  axprlem5OLD  5390  wlkdlem4  29886  lfgriswlk  29889  2pthnloop  29933  eupth2lem3lem4  30435  axprALT2  35409  satfv1lem  35717  wl-3xorfal  37971  sn-axprlem3  42842