Theorem neeqtrri 2855

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 2619	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 2854	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = 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:  cflim2  8968  pnfnemnf  9973  resslem  15760  slotsbhcdif  15903  xrsnsgrp  19601  zlmlem  19684  matbas  20038  matplusg  20039  matvsca  20041  tnglem  22254  uhgrstrrepe  25745  resvlem  29162  limsucncmpi  31614  plusgndxnmulrndx  41743  basendxnmulrndx  41744