Theorem neeqtrri 2999

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 2739	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 2998	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1540   ≠ wne 2926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722  df-ne 2927

This theorem is referenced by:  nlim1  8456  nlim2  8457  1one2o  8613  cflim2  10223  pnfnemnf  11236  basendxnmulrndx  17266  plusgndxnmulrndx  17267  slotsbhcdif  17385  xrsnsgrp  21326  slotsinbpsd  28375  slotslnbpsd  28376  setsvtx  28969  limsucncmpi  36440  sn-1ne2  42260