ifpid2

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

		Ref	Expression
	Assertion	ifpid2	$⊢ φ \leftrightarrow if- (φ, ⊤, ⊥)$

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

Metamath Proof Explorer