Theorem ifpfal 1074

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4468. This is essentially dedlemb 1044. (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 1071	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1073	. 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
3	1, 2	bitrid 282	1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  if-wif 1060

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 397  df-or 845  df-ifp 1061

This theorem is referenced by:  ifpid  1075  elimh  1082  axprlem3  5348  axprlem5  5350  wlkdlem4  28053  lfgriswlk  28056  2pthnloop  28099  eupth2lem3lem4  28595  satfv1lem  33324  wl-3xorfal  35643  sn-axprlem3  40186