Theorem ifpim1 40702

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

Assertion

Ref	Expression
ifpim1	⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))

Proof of Theorem ifpim1

Step	Hyp	Ref	Expression
1		tru 1547	. . . 4 ⊢ ⊤
2	1	olci 866	. . 3 ⊢ (¬ ¬ 𝜑 ∨ ⊤)
3	2	biantrur 534	. 2 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑 ∨ 𝜓)))
4		imor 853	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
5		dfifp4 1067	. 2 ⊢ (if-(¬ 𝜑, ⊤, 𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑 ∨ 𝜓)))
6	3, 4, 5	3bitr4i 306	1 ⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847  if-wif 1063  ⊤wtru 1544

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 848  df-ifp 1064  df-tru 1546

This theorem is referenced by: (None)