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Theorem syl5eqner 3046
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 3039 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792  df-ne 2972
This theorem is referenced by:  xpcoidgend  14057  fclsfnflim  22159  ptcmplem2  22185  vieta1lem1  24406  vieta1lem2  24407  signsvfpn  31182  signsvfnn  31183  finxpreclem2  33725  finxp1o  33727  cdleme3h  36256  cdleme7ga  36269  fourierdlem42  41109
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