ifpnot

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

		Ref	Expression
	Assertion	ifpnot	$⊢ \neg φ \leftrightarrow if- (φ, ⊥, ⊤)$

Step	Hyp	Ref	Expression
1		tru	$⊢ ⊤$
2	1	olci	$⊢ (φ \lor ⊤)$
3	2	biantru	$⊢ (\neg φ \lor ⊥) \leftrightarrow ((\neg φ \lor ⊥) \land (φ \lor ⊤))$
4		fal	$⊢ \neg ⊥$
5	4	biorfi	$⊢ \neg φ \leftrightarrow (\neg φ \lor ⊥)$
6		dfifp4	$⊢ if- (φ, ⊥, ⊤) \leftrightarrow ((\neg φ \lor ⊥) \land (φ \lor ⊤))$
7	3 5 6	3bitr4i	$⊢ \neg φ \leftrightarrow if- (φ, ⊥, ⊤)$

Metamath Proof Explorer