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 2772 . 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 196   = wceq 1631   ∉ wnel 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-clel 2767  df-nel 3047 This theorem is referenced by:  noinfep  8725  wrdlndm  13517  isfbas  21853  upgrreslem  26419  umgrreslem  26420  nbgrnvtx0  26455  nbupgrres  26488  eupth2lem3lem6  27413  frgrncvvdeqlem1  27481  frgrwopreglem4a  27492
 Copyright terms: Public domain W3C validator