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Metamath Proof Explorer

Theorem imim2

Description: A closed form of syllogism (see syl ). Theorem *2.05 of WhiteheadRussell p. 100. Its associated inference is imim2i . Its associated deduction is imim2d . An alternate proof from more basic results is given by ax-1 followed by a2d . (Contributed by NM, 29-Dec-1992) (Proof shortened by Wolf Lammen, 6-Sep-2012)

		Ref	Expression
	Assertion	imim2	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) )
2	1	imim2d	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) )