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Metamath Proof Explorer

Theorem sylnibr

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

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

Proof

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