Theorem 3imtr3g

Description: More general version of 3imtr3i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996) (Proof shortened by Wolf Lammen, 20-Dec-2013)

	Ref	Expression
Hypotheses	3imtr3g.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	3imtr3g.2	⊢ ( 𝜓 ↔ 𝜃 )
	3imtr3g.3	⊢ ( 𝜒 ↔ 𝜏 )
Assertion	3imtr3g	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		3imtr3g.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		3imtr3g.2	⊢ ( 𝜓 ↔ 𝜃 )
3		3imtr3g.3	⊢ ( 𝜒 ↔ 𝜏 )
4	2 1	syl5bir	⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
5	4 3	syl6ib	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )