Theorem neeqtrri 3009

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 2750	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3008	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1548   ≠ wne 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-ne 2937

This theorem is referenced by:  nlim1  8418  nlim2  8419  1one2o  8576  cflim2  10181  pnfnemnf  11196  basendxnmulrndx  17254  plusgndxnmulrndx  17255  slotsbhcdif  17373  xrsnsgrp  21386  slotsinbpsd  28529  slotslnbpsd  28530  setsvtx  29124  limsucncmpi  36686  sn-1ne2  42761