Theorem 3imtr4d

Description: More general version of 3imtr4i . Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995)

	Ref	Expression
Hypotheses	3imtr4d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	3imtr4d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
	3imtr4d.3	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
Assertion	3imtr4d	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		3imtr4d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		3imtr4d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
3		3imtr4d.3	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
4	1 3	sylibrd	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
5	2 4	sylbid	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )