Theorem ifpdfbi

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

	Ref	Expression
Assertion	ifpdfbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		con34b	⊢ ( ( 𝜓 → 𝜑 ) ↔ ( ¬ 𝜑 → ¬ 𝜓 ) )
2	1	anbi2i	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → ¬ 𝜓 ) ) )
3		dfbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
4		dfifp2	⊢ ( if- ( 𝜑 , 𝜓 , ¬ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → ¬ 𝜓 ) ) )
5	2 3 4	3bitr4i	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ¬ 𝜓 ) )