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Theorem raldifb 4075
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 454 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
2 df-nel 3095 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥𝐵)
32anbi2i 625 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4 eldif 3894 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 281 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
65imbi1i 353 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
71, 6bitr3i 280 . 2 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
87ralbii2 3134 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2112  wnel 3094  wral 3109  cdif 3881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nel 3095  df-ral 3114  df-v 3446  df-dif 3887
This theorem is referenced by:  raldifsnb  4692  coprmproddvdslem  15999  poimirlem26  35076  aacllem  45316
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