Theorem ifpnot 42900

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

Assertion

Ref	Expression
ifpnot	⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤))

Proof of Theorem ifpnot

Step	Hyp	Ref	Expression
1		tru 1538	. . . 4 ⊢ ⊤
2	1	olci 865	. . 3 ⊢ (𝜑 ∨ ⊤)
3	2	biantru 529	. 2 ⊢ ((¬ 𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
4		fal 1548	. . 3 ⊢ ¬ ⊥
5	4	biorfi 937	. 2 ⊢ (¬ 𝜑 ↔ (¬ 𝜑 ∨ ⊥))
6		dfifp4 1065	. 2 ⊢ (if-(𝜑, ⊥, ⊤) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
7	3, 5, 6	3bitr4i 303	1 ⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 846  if-wif 1061  ⊤wtru 1535  ⊥wfal 1546

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 847  df-ifp 1062  df-tru 1537  df-fal 1547

This theorem is referenced by: (None)