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Theorem neleq2 2889
 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 2611 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
2 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
31, 2neleq12d 2887 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∉ wnel 2781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-nel 2783 This theorem is referenced by:  noinfep  8440  wrdlndm  13176  isfbas  21443  nbgra0nb  25958  cusgrares  26001  frgrawopreglem4  26574  nbgrnvtx0  40563  nbupgrres  40592  eupth2lem3lem6  41401  frgrwopreglem4  41484
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