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Theorem dfifp6 1038
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 1033 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 ancom 465 . . . 4 ((¬ 𝜑𝜒) ↔ (𝜒 ∧ ¬ 𝜑))
3 annim 440 . . . 4 ((𝜒 ∧ ¬ 𝜑) ↔ ¬ (𝜒𝜑))
42, 3bitri 264 . . 3 ((¬ 𝜑𝜒) ↔ ¬ (𝜒𝜑))
54orbi2i 540 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
61, 5bitri 264 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  if-wif 1032
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 384  df-an 385  df-ifp 1033
This theorem is referenced by:  dfifp7  1039  ifpdfan2  38124
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