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Theorem dfifp6 1064
 Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
Assertion
Ref Expression
dfifp6 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))

Proof of Theorem dfifp6
StepHypRef Expression
1 df-ifp 1059 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 ancom 464 . . . 4 ((¬ 𝜑𝜒) ↔ (𝜒 ∧ ¬ 𝜑))
3 annim 407 . . . 4 ((𝜒 ∧ ¬ 𝜑) ↔ ¬ (𝜒𝜑))
42, 3bitri 278 . . 3 ((¬ 𝜑𝜒) ↔ ¬ (𝜒𝜑))
54orbi2i 910 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
61, 5bitri 278 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058 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 400  df-or 845  df-ifp 1059 This theorem is referenced by:  dfifp7  1065  ifpdfan2  40186
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