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	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
syl5eqner	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem syl5eqner

Step	Hyp	Ref	Expression
1		syl5eqner.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = 𝐴)
3		syl5eqner.2	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
4	2, 3	eqnetrrd 3039	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

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