Theorem ralimralim 44972

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 3290	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
2		rspa 3254	. . . 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 412	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
6	1, 5	ralrimi 3263	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-ral 3068

This theorem is referenced by:  infxrunb2  45272