MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neleq2 Structured version   Visualization version   GIF version

Theorem neleq2 3052
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 2734 . 2 (𝐴 = 𝐵𝐶 = 𝐶)
2 id 22 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
31, 2neleq12d 3050 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wnel 3046
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  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-clel 2811  df-nel 3047
This theorem is referenced by:  noinfep  9601  isfbas  23196  upgrreslem  28294  umgrreslem  28295  nbgrnvtx0  28329  nbupgrres  28354  eupth2lem3lem6  29219  frgrncvvdeqlem1  29285  frgrwopreglem4a  29296
  Copyright terms: Public domain W3C validator