Theorem ralin 4196

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 3913	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	imbi1i 349	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑))
3		impexp 450	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
4	2, 3	bitri 275	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)))
5	4	ralbii2 3074	1 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047   ∩ cin 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-in 3904

This theorem is referenced by:  ref5  38361  sswfaxreg  45090