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

This proof depends on syntax axioms:   = wceq 1569   ≠ wne 2958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2755  df-ne 2959

This theorem is used by:  nlim1  8508  nlim2  8509  1one2o  8666  pnfnemnf  11374  basendxnmulrndx  17479  plusgndxnmulrndx  17480  slotsbhcdif  17598  xrsnsgrp  21726  slotsinbpsd  28915  slotslnbpsd  28916  setsvtx  29625  kard0b  35898  limsucncmpi  37360  sn-1ne2  43470  fuconuniqlem  50920