Metamath Proof Explorer

Theorem ifpdfbi

Description: Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020) (Proof shortened by Wolf Lammen, 30-Apr-2024) (Proof shortened by Garrett Katz, 25-Jun-2026)

		Ref	Expression
	Assertion	ifpdfbi	$⊢ (φ \leftrightarrow ψ) \leftrightarrow if- (φ, ψ, \neg ψ)$

Proof

Step	Hyp	Ref	Expression
1		dfbi3	$⊢ (φ \leftrightarrow ψ) \leftrightarrow ((φ \land ψ) \lor (\neg φ \land \neg ψ))$
2		df-ifp	$⊢ if- (φ, ψ, \neg ψ) \leftrightarrow ((φ \land ψ) \lor (\neg φ \land \neg ψ))$
3	1 2	bitr4i	$⊢ (φ \leftrightarrow ψ) \leftrightarrow if- (φ, ψ, \neg ψ)$