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Theorem nelbOLD 3222
Description: Obsolete version of nelb 3221 as of 3-Nov-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nelbOLD 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelbOLD
StepHypRef Expression
1 df-ne 2941 . . . . 5 (𝑥𝐴 ↔ ¬ 𝑥 = 𝐴)
21ralbii 3093 . . . 4 (∀𝑥𝐵 𝑥𝐴 ↔ ∀𝑥𝐵 ¬ 𝑥 = 𝐴)
3 ralnex 3072 . . . 4 (∀𝑥𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥𝐵 𝑥 = 𝐴)
42, 3bitri 275 . . 3 (∀𝑥𝐵 𝑥𝐴 ↔ ¬ ∃𝑥𝐵 𝑥 = 𝐴)
5 risset 3220 . . 3 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
64, 5xchbinxr 335 . 2 (∀𝑥𝐵 𝑥𝐴 ↔ ¬ 𝐴𝐵)
76bicomi 223 1 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070
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 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071
This theorem is referenced by: (None)
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