Theorem dfbi1

Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992)

	Ref	Expression
Assertion	dfbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-bi	⊢ ¬ ( ( ( 𝜑 ↔ 𝜓 ) → ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) → ¬ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) )
2		impbi	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) → ( ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) ) )
3	2	con3rr3	⊢ ( ¬ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) → ( ( ( 𝜑 ↔ 𝜓 ) → ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) → ¬ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) ) )
4	1 3	mt3	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) )