Theorem neeqtrri 3005

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 2745	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3004	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1541   ≠ wne 2932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2728  df-ne 2933

This theorem is referenced by:  nlim1  8416  nlim2  8417  1one2o  8574  cflim2  10173  pnfnemnf  11187  basendxnmulrndx  17216  plusgndxnmulrndx  17217  slotsbhcdif  17335  xrsnsgrp  21362  slotsinbpsd  28513  slotslnbpsd  28514  setsvtx  29108  limsucncmpi  36639  sn-1ne2  42520