Theorem bj-alsyl

Description: Syllogism under the universal quantifier, in the curried form appearing as Theorem *10.3 of WhiteheadRussell p. 145. See alsyl for the uncurried form. (Contributed by BJ, 28-Mar-2026)

	Ref	Expression
Assertion	bj-alsyl	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \forall x\left(\phi \to \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imim1	$$⊢\left(\phi \to \psi \right)\to \left(\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)\right)$$
2	1	al2imi	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \forall x\left(\phi \to \chi \right)\right)$$