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	⊢ (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 2891	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

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