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	$$⊢\phi \to \left(\psi \to \chi \right)$$
	syld.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
Assertion	syld	$$⊢\phi \to \left(\psi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		syld.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		syld.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
3	2	a1d	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
4	1 3	mpdd	$$⊢\phi \to \left(\psi \to \theta \right)$$