Theorem ifpfal 1090

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4501. This is essentially dedlemb 1060. (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 1088	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1089	. 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
3	1, 2	bitrid 286	1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  if-wif 1076

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 210  df-an 401  df-or 861  df-ifp 1077

This theorem is referenced by:  ifpid  1091  elimh  1097  axprlem3  5397  axpr  5399  axprlem3OLD  5401  axprlem5OLD  5403  wlkdlem4  29974  lfgriswlk  29977  2pthnloop  30021  eupth2lem3lem4  30523  axprALT2  35446  satfv1lem  35787  wl-3xorfal  38040  sn-axprlem3  42913