MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl5eqner Structured version   Visualization version   GIF version

Theorem syl5eqner 2898
Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
Hypotheses
Ref Expression
syl5eqner.1 𝐵 = 𝐴
syl5eqner.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
syl5eqner (𝜑𝐴𝐶)

Proof of Theorem syl5eqner
StepHypRef Expression
1 syl5eqner.1 . . 3 𝐵 = 𝐴
21a1i 11 . 2 (𝜑𝐵 = 𝐴)
3 syl5eqner.2 . 2 (𝜑𝐵𝐶)
42, 3eqnetrrd 2891 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wne 2823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644  df-ne 2824
This theorem is referenced by:  xpcoidgend  13760  fclsfnflim  21878  ptcmplem2  21904  vieta1lem1  24110  vieta1lem2  24111  signsvfpn  30790  signsvfnn  30791  finxpreclem2  33357  finxp1o  33359  cdleme3h  35840  cdleme7ga  35853  fourierdlem42  40684
  Copyright terms: Public domain W3C validator