Theorem dfifp6 1065

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)

Assertion

Ref	Expression
dfifp6	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))

Proof of Theorem dfifp6

Step	Hyp	Ref	Expression
1		df-ifp 1060	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		ancom 460	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝜒) ↔ (𝜒 ∧ ¬ 𝜑))
3		annim 403	. . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) ↔ ¬ (𝜒 → 𝜑))
4	2, 3	bitri 274	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) ↔ ¬ (𝜒 → 𝜑))
5	4	orbi2i 909	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))
6	1, 5	bitri 274	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059

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 396  df-or 844  df-ifp 1060

This theorem is referenced by:  dfifp7  1066  ifpdfan2  40968