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Theorem syl5eqner 2857
 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 2850 1 (𝜑𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-cleq 2603  df-ne 2782 This theorem is referenced by:  xpcoidgend  13562  fclsfnflim  21641  ptcmplem2  21667  vieta1lem1  23869  vieta1lem2  23870  signsvfpn  29988  signsvfnn  29989  finxpreclem2  32403  finxp1o  32405  cdleme3h  34540  cdleme7ga  34553  fourierdlem42  39042
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