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Theorem raldifb 4079
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 451 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
2 df-nel 3050 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥𝐵)
32anbi2i 623 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4 eldif 3897 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 277 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
65imbi1i 350 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
71, 6bitr3i 276 . 2 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
87ralbii2 3089 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2106  wnel 3049  wral 3064  cdif 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nel 3050  df-ral 3069  df-v 3432  df-dif 3890
This theorem is referenced by:  raldifsnb  4730  coprmproddvdslem  16355  poimirlem26  35789  aacllem  46461
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