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Metamath Proof Explorer

Theorem syl6d

Description: A nested syllogism deduction. Deduction associated with syl6 . (Contributed by NM, 11-May-1993) (Proof shortened by Josh Purinton, 29-Dec-2000) (Proof shortened by Mel L. O'Cat, 2-Feb-2006)

	Ref	Expression
Hypotheses	syl6d.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
	syl6d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
Assertion	syl6d	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		syl6d.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
2		syl6d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
3	2	a1d	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜏 ) ) )
4	1 3	syldd	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )