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Metamath Proof Explorer

Theorem imim12d

Description: Deduction combining antecedents and consequents. Deduction associated with imim12 and imim12i . (Contributed by NM, 7-Aug-1994) (Proof shortened by Mel L. O'Cat, 30-Oct-2011)

	Ref	Expression
Hypotheses	imim12d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	imim12d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
Assertion	imim12d	⊢ ( 𝜑 → ( ( 𝜒 → 𝜃 ) → ( 𝜓 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imim12d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		imim12d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
3	2	imim2d	⊢ ( 𝜑 → ( ( 𝜒 → 𝜃 ) → ( 𝜒 → 𝜏 ) ) )
4	1 3	syl5d	⊢ ( 𝜑 → ( ( 𝜒 → 𝜃 ) → ( 𝜓 → 𝜏 ) ) )