Theorem ifpid3g 41099

Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpid3g	⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓)))

Proof of Theorem ifpid3g

Step	Hyp	Ref	Expression
1		olc 865	. . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
2	1, 1	pm3.2i 471	. 2 ⊢ ((𝜒 → (𝜑 ∨ 𝜒)) ∧ (𝜒 → (𝜑 ∨ 𝜒)))
3		ifpidg 41098	. 2 ⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓)) ∧ ((𝜒 → (𝜑 ∨ 𝜒)) ∧ (𝜒 → (𝜑 ∨ 𝜒)))))
4	2, 3	mpbiran2 707	1 ⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  if-wif 1060

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 397  df-or 845  df-ifp 1061

This theorem is referenced by: (None)