Theorem ifpfal 1024

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4095. This is essentially dedlemb 1003. (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 1022	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
2		ifptru 1023	. 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒))
3	1, 2	syl5bb 272	1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpid  1025  elimh  1030  wlkdlem4  26582  lfgriswlk  26585  2pthnloop  26627  eupth2lem3lem4  27091