Theorem syl5eqner 2869

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 2862	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1483   ≠ wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-ne 2795

This theorem is referenced by:  xpcoidgend  13714  fclsfnflim  21831  ptcmplem2  21857  vieta1lem1  24065  vieta1lem2  24066  signsvfpn  30662  signsvfnn  30663  finxpreclem2  33227  finxp1o  33229  cdleme3h  35522  cdleme7ga  35535  fourierdlem42  40366