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Metamath Proof Explorer

Theorem sylnbi

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

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

Proof

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