Theorem ralin 4201

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 3920	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	imbi1i 351	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑))
3		impexp 454	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
4	2, 3	bitri 277	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
5	4	ralbii2 3104	1 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  ∀wral 3076   ∩ cin 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-v 3456  df-in 3911

This theorem is referenced by:  mh-infprim2bi  36907  ref5  38818  sswfaxreg  45563