Theorem neeqtrri 3028

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 2769	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3027	1 ⊢ 𝐴 ≠ 𝐶

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-ne 2956

This theorem is used by:  nlim1  8479  nlim2  8480  1one2o  8637  cflim2  10268  pnfnemnf  11291  basendxnmulrndx  17384  plusgndxnmulrndx  17385  slotsbhcdif  17503  xrsnsgrp  21624  slotsinbpsd  28785  slotslnbpsd  28786  setsvtx  29495  kard0b  35688  limsucncmpi  37067  sn-1ne2  43149