Theorem neeqtrri 3005

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 2744	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3004	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1540   ≠ wne 2932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-ne 2933

This theorem is referenced by:  nlim1  8499  nlim2  8500  1one2o  8656  cflim2  10275  pnfnemnf  11288  basendxnmulrndx  17308  plusgndxnmulrndx  17309  slotsbhcdif  17427  xrsnsgrp  21368  slotsinbpsd  28366  slotslnbpsd  28367  setsvtx  28960  limsucncmpi  36409  sn-1ne2  42262