Theorem ralimralim 44511

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 3272	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
2		rspa 3236	. . . 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 411	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
6	1, 5	ralrimi 3245	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-ral 3052

This theorem is referenced by:  infxrunb2  44812