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	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ∀ 𝑥 ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imim1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2	1	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ∀ 𝑥 ( 𝜑 → 𝜒 ) ) )