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Theorem dfdif3 4042
 Description: Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
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 3890 . 2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
2 ax6ev 1972 . . . . . . 7 𝑦 𝑦 = 𝑥
32biantrur 534 . . . . . 6 𝑥𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
4 19.41v 1950 . . . . . 6 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 281 . . . . 5 𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
6 sb56 2274 . . . . 5 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵))
7 equcom 2025 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
87imbi1i 353 . . . . . . 7 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥𝐵))
9 eleq1w 2872 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
109notbid 321 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝐵 ↔ ¬ 𝑦𝐵))
1110pm5.74i 274 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦𝐵))
12 con2b 363 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑦𝐵) ↔ (𝑦𝐵 → ¬ 𝑥 = 𝑦))
13 df-ne 2988 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
1413bicomi 227 . . . . . . . . 9 𝑥 = 𝑦𝑥𝑦)
1514imbi2i 339 . . . . . . . 8 ((𝑦𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦𝐵𝑥𝑦))
1611, 12, 153bitri 300 . . . . . . 7 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
178, 16bitri 278 . . . . . 6 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
1817albii 1821 . . . . 5 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
195, 6, 183bitri 300 . . . 4 𝑥𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
20 df-ral 3111 . . . 4 (∀𝑦𝐵 𝑥𝑦 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
2119, 20bitr4i 281 . . 3 𝑥𝐵 ↔ ∀𝑦𝐵 𝑥𝑦)
2221rabbii 3420 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
231, 22eqtri 2821 1 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110   ∖ cdif 3878 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 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rab 3115  df-dif 3884 This theorem is referenced by: (None)
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