Theorem ifpnot 44237

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 1574	. . . 4 ⊢ ⊤
2	1	olci 880	. . 3 ⊢ (𝜑 ∨ ⊤)
3	2	biantru 539	. 2 ⊢ ((¬ 𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
4		fal 1584	. . 3 ⊢ ¬ ⊥
5	4	biorfri 953	. 2 ⊢ (¬ 𝜑 ↔ (¬ 𝜑 ∨ ⊥))
6		dfifp4 1082	. 2 ⊢ (if-(𝜑, ⊥, ⊤) ↔ ((¬ 𝜑 ∨ ⊥) ∧ (𝜑 ∨ ⊤)))
7	3, 5, 6	3bitr4i 306	1 ⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861  if-wif 1078  ⊤wtru 1571  ⊥wfal 1582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-tru 1573  df-fal 1583

This theorem is used by: (None)