Theorem sylnbir

Description: A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013)

	Ref	Expression
Hypotheses	sylnbir.1	⊢ ( 𝜓 ↔ 𝜑 )
	sylnbir.2	⊢ ( ¬ 𝜓 → 𝜒 )
Assertion	sylnbir	⊢ ( ¬ 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		sylnbir.1	⊢ ( 𝜓 ↔ 𝜑 )
2		sylnbir.2	⊢ ( ¬ 𝜓 → 𝜒 )
3	1	bicomi	⊢ ( 𝜑 ↔ 𝜓 )
4	3 2	sylnbi	⊢ ( ¬ 𝜑 → 𝜒 )