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Theorem neleq2 3081
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Assertion
Ref Expression
neleq2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem neleq2
StepHypRef Expression
1 eqidd 2781 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
2 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
31, 2neleq12d 3079 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1508  wnel 3075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2752
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-cleq 2773  df-clel 2848  df-nel 3076
This theorem is referenced by:  noinfep  8923  wrdlndmOLD  13697  isfbas  22156  upgrreslem  26804  umgrreslem  26805  nbgrnvtx0  26839  nbupgrres  26864  eupth2lem3lem6  27778  frgrncvvdeqlem1  27848  frgrwopreglem4a  27859
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