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Theorem ralin 38229
Description: Restricted universal quantification over intersection. (Contributed by Peter Mazsa, 8-Sep-2023.)
Assertion
Ref Expression
ralin (∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∀𝑥𝐴 (𝑥𝐵𝜑))

Proof of Theorem ralin
StepHypRef Expression
1 elin 3978 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21imbi1i 349 . . 3 ((𝑥 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝑥𝐴𝑥𝐵) → 𝜑))
3 impexp 450 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
42, 3bitri 275 . 2 ((𝑥 ∈ (𝐴𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
54ralbii2 3086 1 (∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∀𝑥𝐴 (𝑥𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  wral 3058  cin 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-v 3479  df-in 3969
This theorem is referenced by:  ref5  38294
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