MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelne2OLD Structured version   Visualization version   GIF version

Theorem nelne2OLD 3113
Description: Obsolete version of nelne2 3112 asw of 14-May-2023. (Contributed by NM, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nelne2OLD ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Proof of Theorem nelne2OLD
StepHypRef Expression
1 eleq1 2897 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 250 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 3027 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 407 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-ne 3014
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator