Theorem imim12

Description: Closed form of imim12i and of 3syl . (Contributed by BJ, 16-Jul-2019)

	Ref	Expression
Assertion	imim12	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		imim2	⊢ ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) )
2		imim1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜃 ) ) )
3	1 2	syl9r	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜃 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜃 ) ) ) )