Theorem alsyl

Description: Universally quantified and uncurried (imported) form of syllogism. Theorem *10.3 in WhiteheadRussell p. 150. (Contributed by Andrew Salmon, 8-Jun-2011)

	Ref	Expression
Assertion	alsyl	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜓 → 𝜒 ) ) → ∀ 𝑥 ( 𝜑 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		pm3.33	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) )
2	1	alanimi	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜓 → 𝜒 ) ) → ∀ 𝑥 ( 𝜑 → 𝜒 ) )