Theorem 3imtr3d

Description: More general version of 3imtr3i . Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996)

	Ref	Expression
Hypotheses	3imtr3d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	3imtr3d.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
	3imtr3d.3	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
Assertion	3imtr3d	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		3imtr3d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		3imtr3d.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
3		3imtr3d.3	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
4	1 3	sylibd	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
5	2 4	sylbird	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )