Theorem neeqtrri 2854

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
neeqtrr.1	⊢ 𝐴 ≠ 𝐵
neeqtrr.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
neeqtrri	⊢ 𝐴 ≠ 𝐶

Proof of Theorem neeqtrri

Step	Hyp	Ref	Expression
1		neeqtrr.1	. 2 ⊢ 𝐴 ≠ 𝐵
2		neeqtrr.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2618	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 2853	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1474   ≠ wne 2779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-ext 2589

This theorem depends on definitions:  df-bi 195  df-cleq 2602  df-ne 2781

This theorem is referenced by:  cflim2  8945  pnfnemnf  11786  resslem  15706  slotsbhcdif  15849  xrsnsgrp  19547  zlmlem  19629  matbas  19980  matplusg  19981  matvsca  19983  tnglem  22192  resvlem  28968  limsucncmpi  31420  uhgrstrrepe  40299  plusgndxnmulrndx  41738  basendxnmulrndx  41739