Theorem neeqtrri 3037

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 2778	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3036	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1567   ≠ wne 2964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-ne 2965

This theorem is referenced by:  nlim1  8473  nlim2  8474  1one2o  8631  cflim2  10246  pnfnemnf  11263  basendxnmulrndx  17348  plusgndxnmulrndx  17349  slotsbhcdif  17467  xrsnsgrp  21526  slotsinbpsd  28675  slotslnbpsd  28676  setsvtx  29325  limsucncmpi  36844  sn-1ne2  42921