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 ↔ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi3	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) )
2		df-ifp	⊢ ( if- ( 𝜑 , 𝜓 , ¬ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) )
3	1 2	bitr4i	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜓 ) )