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Theorem neleq2 3097
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 2799 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
2 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
31, 2neleq12d 3095 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wnel 3091
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  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-nel 3092
This theorem is referenced by:  noinfep  9107  isfbas  22434  upgrreslem  27094  umgrreslem  27095  nbgrnvtx0  27129  nbupgrres  27154  eupth2lem3lem6  28018  frgrncvvdeqlem1  28084  frgrwopreglem4a  28095
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