Theorem biimpr

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999) (Proof shortened by Wolf Lammen, 11-Nov-2012)

	Ref	Expression
Assertion	biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) )
2		simprim	⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜓 → 𝜑 ) )
3	1 2	sylbi	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )