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Theorem nelb 3223
Description: A definition of ¬ 𝐴𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
Assertion
Ref Expression
nelb 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelb
StepHypRef Expression
1 df-ne 2933 . . . 4 (𝑥𝐴 ↔ ¬ 𝑥 = 𝐴)
21ralbii 3085 . . 3 (∀𝑥𝐵 𝑥𝐴 ↔ ∀𝑥𝐵 ¬ 𝑥 = 𝐴)
3 ralnex 3064 . . 3 (∀𝑥𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥𝐵 𝑥 = 𝐴)
42, 3bitr2i 276 . 2 (¬ ∃𝑥𝐵 𝑥 = 𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
5 risset 3222 . 2 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
64, 5xchnxbir 333 1 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  wne 2932  wral 3053  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063
This theorem is referenced by:  inpr0  32241
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