Theorem neeqtrri 3032

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 3031	1 ⊢ 𝐴 ≠ 𝐶

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2959

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ne 2960

This theorem is used by:  nlim1  8535  nlim2  8536  1one2o  8693  pnfnemnf  11403  basendxnmulrndx  17513  plusgndxnmulrndx  17514  slotsbhcdif  17636  xrsnsgrp  21766  slotsinbpsd  28960  slotslnbpsd  28961  setsvtx  29670  kard0b  35968  limsucncmpi  37448  sn-1ne2  43596  fuconuniqlem  51112