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Metamath Proof Explorer

Theorem sylibr

Description: A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993)

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

Proof

Step	Hyp	Ref	Expression
1		sylibr.1	⊢ ( 𝜑 → 𝜓 )
2		sylibr.2	⊢ ( 𝜒 ↔ 𝜓 )
3	2	biimpri	⊢ ( 𝜓 → 𝜒 )
4	1 3	syl	⊢ ( 𝜑 → 𝜒 )