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Theorem dfdif3 4126
Description: Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof shortened by SN, 15-Aug-2025.)
Assertion
Ref Expression
dfdif3 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfdif3
StepHypRef Expression
1 dfdif2 3971 . 2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
2 nelb 3231 . . 3 𝑥𝐵 ↔ ∀𝑦𝐵 𝑦𝑥)
3 necom 2991 . . . 4 (𝑦𝑥𝑥𝑦)
43ralbii 3090 . . 3 (∀𝑦𝐵 𝑦𝑥 ↔ ∀𝑦𝐵 𝑥𝑦)
52, 4bitri 275 . 2 𝑥𝐵 ↔ ∀𝑦𝐵 𝑥𝑦)
61, 5rabbieq 3441 1 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  cdif 3959
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-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-dif 3965
This theorem is referenced by: (None)
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