Theorem bj-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) State the most general derivable instance. (Revised by BJ, 5-Apr-2026)

	Ref	Expression
Hypotheses	bj-alrimdh.nf1	$$⊢\phi \to \forall x\psi$$
	bj-alrimdh.nf2	$$⊢\chi \to \forall x\theta$$
	bj-alrimdh.maj	$$⊢\psi \to \left(\theta \to \tau \right)$$
Assertion	bj-alrimdh	$$⊢\phi \to \left(\chi \to \forall x\tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-alrimdh.nf1	$$⊢\phi \to \forall x\psi$$
2		bj-alrimdh.nf2	$$⊢\chi \to \forall x\theta$$
3		bj-alrimdh.maj	$$⊢\psi \to \left(\theta \to \tau \right)$$
4	1 3	bj-alimdh	$$⊢\phi \to \left(\forall x\theta \to \forall x\tau \right)$$
5	2 4	syl5	$$⊢\phi \to \left(\chi \to \forall x\tau \right)$$