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Theorem raldifb 3783
Description: Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
Assertion
Ref Expression
raldifb (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)

Proof of Theorem raldifb
StepHypRef Expression
1 impexp 461 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
2 df-nel 2927 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥𝐵)
32anbi2i 730 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4 eldif 3617 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 267 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
65imbi1i 338 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
71, 6bitr3i 266 . 2 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
87ralbii2 3007 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wcel 2030  wnel 2926  wral 2941  cdif 3604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-ral 2946  df-v 3233  df-dif 3610
This theorem is referenced by:  raldifsnb  4358  coprmproddvdslem  15423  poimirlem26  33565  aacllem  42875
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