Theorem ifpfal 1092

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4496. This is essentially dedlemb 1062. (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 1090	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1091	. 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
3	1, 2	bitrid 286	1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079

This theorem is used by:  ifpid  1093  elimh  1099  axprlem3  5396  axpr  5398  axprlem3OLD  5400  axprlem5OLD  5402  wlkdlem4  30042  lfgriswlk  30045  2pthnloop  30089  eupth2lem3lem4  30591  axprALT2  35512  satfv1lem  35862  wl-3xorfal  38146  sn-axprlem3  43017