Theorem ralim

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997) (Proof shortened by Wolf Lammen, 1-Dec-2019)

	Ref	Expression
Assertion	ralim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) )
2	1	ral2imi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )