Theorem ifpfal 1082

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4466. This is essentially dedlemb 1053. (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 1080	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1081	. 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 1069

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 398  df-or 855  df-ifp 1070

This theorem is referenced by:  ifpid  1083  elimh  1089  axprlem3  5357  axpr  5359  axprlem3OLD  5361  axprlem5OLD  5363  wlkdlem4  29774  lfgriswlk  29777  2pthnloop  29821  eupth2lem3lem4  30323  axprALT2  35305  satfv1lem  35605  wl-3xorfal  37849  sn-axprlem3  42720