Theorem neeqtrri 3031

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 2772	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3030	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1570   ≠ wne 2958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-ne 2959

This theorem is referenced by:  nlim1  8470  nlim2  8471  1one2o  8628  cflim2  10242  pnfnemnf  11259  basendxnmulrndx  17344  plusgndxnmulrndx  17345  slotsbhcdif  17463  xrsnsgrp  21558  slotsinbpsd  28710  slotslnbpsd  28711  setsvtx  29385  kard0b  35572  limsucncmpi  36976  sn-1ne2  43052