Theorem 3imtr4i

Description: A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993)

	Ref	Expression
Hypotheses	3imtr4.1	⊢ ( 𝜑 → 𝜓 )
	3imtr4.2	⊢ ( 𝜒 ↔ 𝜑 )
	3imtr4.3	⊢ ( 𝜃 ↔ 𝜓 )
Assertion	3imtr4i	⊢ ( 𝜒 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		3imtr4.1	⊢ ( 𝜑 → 𝜓 )
2		3imtr4.2	⊢ ( 𝜒 ↔ 𝜑 )
3		3imtr4.3	⊢ ( 𝜃 ↔ 𝜓 )
4	2 1	sylbi	⊢ ( 𝜒 → 𝜓 )
5	4 3	sylibr	⊢ ( 𝜒 → 𝜃 )