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Metamath Proof Explorer

Theorem imim12i

Description: Inference joining two implications. Inference associated with imim12 . Its associated inference is 3syl . (Contributed by NM, 12-Mar-1993) (Proof shortened by Mel L. O'Cat, 29-Oct-2011)

	Ref	Expression
Hypotheses	imim12i.1	⊢ ( 𝜑 → 𝜓 )
	imim12i.2	⊢ ( 𝜒 → 𝜃 )
Assertion	imim12i	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		imim12i.1	⊢ ( 𝜑 → 𝜓 )
2		imim12i.2	⊢ ( 𝜒 → 𝜃 )
3	2	imim2i	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜓 → 𝜃 ) )
4	1 3	syl5	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜃 ) )