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Metamath Proof Explorer

Theorem sylbi

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

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

Proof

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