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Theorem nelbOLD 30338
 Description: Obsolete version of nelb 3192 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
nelbOLD 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelbOLD
StepHypRef Expression
1 df-ral 3075 . 2 (∀𝑥𝐵 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(𝑥𝐵 → ¬ 𝑥 = 𝐴))
2 df-ne 2952 . . 3 (𝑥𝐴 ↔ ¬ 𝑥 = 𝐴)
32ralbii 3097 . 2 (∀𝑥𝐵 𝑥𝐴 ↔ ∀𝑥𝐵 ¬ 𝑥 = 𝐴)
4 dfclel 2831 . . . 4 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
54notbii 323 . . 3 𝐴𝐵 ↔ ¬ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
6 alnex 1783 . . 3 (∀𝑥 ¬ (𝑥 = 𝐴𝑥𝐵) ↔ ¬ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
7 imnan 403 . . . . 5 ((𝑥𝐵 → ¬ 𝑥 = 𝐴) ↔ ¬ (𝑥𝐵𝑥 = 𝐴))
8 ancom 464 . . . . . 6 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝑥𝐵))
98notbii 323 . . . . 5 (¬ (𝑥𝐵𝑥 = 𝐴) ↔ ¬ (𝑥 = 𝐴𝑥𝐵))
107, 9bitr2i 279 . . . 4 (¬ (𝑥 = 𝐴𝑥𝐵) ↔ (𝑥𝐵 → ¬ 𝑥 = 𝐴))
1110albii 1821 . . 3 (∀𝑥 ¬ (𝑥 = 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐵 → ¬ 𝑥 = 𝐴))
125, 6, 113bitr2i 302 . 2 𝐴𝐵 ↔ ∀𝑥(𝑥𝐵 → ¬ 𝑥 = 𝐴))
131, 3, 123bitr4ri 307 1 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2830  df-ne 2952  df-ral 3075 This theorem is referenced by: (None)
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