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Metamath Proof Explorer

Theorem syld

Description: Syllogism deduction. Deduction associated with syl . See conventions for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993) (Proof shortened by Mel L. O'Cat, 19-Feb-2008) (Proof shortened by Wolf Lammen, 3-Aug-2012)

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

Proof

Step	Hyp	Ref	Expression
1		syld.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syld.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3	2	a1d	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
4	1 3	mpdd	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )