Theorem neeqtrri 3030

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 2771	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3029	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1560   ≠ wne 2957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-cleq 2754  df-ne 2958

This theorem is referenced by:  nlim1  8458  nlim2  8459  1one2o  8616  cflim2  10220  pnfnemnf  11237  basendxnmulrndx  17325  plusgndxnmulrndx  17326  slotsbhcdif  17444  xrsnsgrp  21460  slotsinbpsd  28610  slotslnbpsd  28611  setsvtx  29236  limsucncmpi  36805  sn-1ne2  42880