Theorem syl6bir

Description: A mixed syllogism inference. (Contributed by NM, 18-May-1994)

	Ref	Expression
Hypotheses	syl6bir.1	$$⊢\phi \to \left(\chi \leftrightarrow \psi \right)$$
	syl6bir.2	$$⊢\chi \to \theta$$
Assertion	syl6bir	$$⊢\phi \to \left(\psi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		syl6bir.1	$$⊢\phi \to \left(\chi \leftrightarrow \psi \right)$$
2		syl6bir.2	$$⊢\chi \to \theta$$
3	1	biimprd	$$⊢\phi \to \left(\psi \to \chi \right)$$
4	3 2	syl6	$$⊢\phi \to \left(\psi \to \theta \right)$$