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	⊢ (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 2850	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

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