Theorem 3imtr3i

Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994)

	Ref	Expression
Hypotheses	3imtr3.1	$$⊢\phi \to \psi$$
	3imtr3.2	$$⊢\phi \leftrightarrow \chi$$
	3imtr3.3	$$⊢\psi \leftrightarrow \theta$$
Assertion	3imtr3i	$$⊢\chi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		3imtr3.1	$$⊢\phi \to \psi$$
2		3imtr3.2	$$⊢\phi \leftrightarrow \chi$$
3		3imtr3.3	$$⊢\psi \leftrightarrow \theta$$
4	2 1	sylbir	$$⊢\chi \to \psi$$
5	4 3	sylib	$$⊢\chi \to \theta$$