Theorem alrimdh

Description: Deduction form of Theorem 19.21 of Margaris p. 90, see 19.21 and 19.21h . (Contributed by NM, 10-Feb-1997) (Proof shortened by Andrew Salmon, 13-May-2011)

	Ref	Expression
Hypotheses	alrimdh.1	$$⊢\phi \to \forall x\phi$$
	alrimdh.2	$$⊢\psi \to \forall x\psi$$
	alrimdh.3	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	alrimdh	$$⊢\phi \to \left(\psi \to \forall x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		alrimdh.1	$$⊢\phi \to \forall x\phi$$
2		alrimdh.2	$$⊢\psi \to \forall x\psi$$
3		alrimdh.3	$$⊢\phi \to \left(\psi \to \chi \right)$$
4	1 3	alimdh	$$⊢\phi \to \left(\forall x\psi \to \forall x\chi \right)$$
5	2 4	syl5	$$⊢\phi \to \left(\psi \to \forall x\chi \right)$$