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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	$⊢ φ \leftrightarrow ψ$
	sylbi.2	$⊢ ψ \to χ$
Assertion	sylbi	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		sylbi.1	$⊢ φ \leftrightarrow ψ$
2		sylbi.2	$⊢ ψ \to χ$
3	1	biimpi	$⊢ φ \to ψ$
4	3 2	syl	$⊢ φ \to χ$