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Theorem ralin 37713
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 3961 . . . 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 3085 1 (∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∀𝑥𝐴 (𝑥𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2099  wral 3057  cin 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-v 3472  df-in 3952
This theorem is referenced by:  ref5  37779
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