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	⊢ ( 𝜑 → 𝜓 )
	3imtr3.2	⊢ ( 𝜑 ↔ 𝜒 )
	3imtr3.3	⊢ ( 𝜓 ↔ 𝜃 )
Assertion	3imtr3i	⊢ ( 𝜒 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		3imtr3.1	⊢ ( 𝜑 → 𝜓 )
2		3imtr3.2	⊢ ( 𝜑 ↔ 𝜒 )
3		3imtr3.3	⊢ ( 𝜓 ↔ 𝜃 )
4	2 1	sylbir	⊢ ( 𝜒 → 𝜓 )
5	4 3	sylib	⊢ ( 𝜒 → 𝜃 )