Theorem ralin 4210

Description: Restricted universal quantification over intersection. (Contributed by Peter Mazsa, 8-Sep-2023.)

Assertion

Ref	Expression
ralin	⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))

Proof of Theorem ralin

Step	Hyp	Ref	Expression
1		elin 3929	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	imbi1i 352	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑))
3		impexp 455	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
4	2, 3	bitri 278	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
5	4	ralbii2 3113	1 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ∀wral 3085   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3465  df-in 3920

This theorem is referenced by:  mh-infprim2bi  36947  ref5  38858  sswfaxreg  45588