Theorem ralimralim 38279

Description: Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

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

Proof of Theorem ralimralim

Step	Hyp	Ref	Expression
1		nfra1 2925	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
2		rspa 2914	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜑)
3		ax-1 6	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜑))
4	2, 3	syl 17	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑))
5	4	ex 449	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
6	1, 5	ralrimi 2940	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-ral 2901

This theorem is referenced by:  infxrunb2  38525