Theorem dfbi2

Description: A theorem similar to the standard definition of the biconditional. Definition of Margaris p. 49. (Contributed by NM, 24-Jan-1993)

	Ref	Expression
Assertion	dfbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) )
2		df-an	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) )
3	1 2	bitr4i	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )