Theorem neeqtrri 3001

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 2740	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3000	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1541   ≠ wne 2928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-ne 2929

This theorem is referenced by:  nlim1  8404  nlim2  8405  1one2o  8561  cflim2  10154  pnfnemnf  11167  basendxnmulrndx  17200  plusgndxnmulrndx  17201  slotsbhcdif  17319  xrsnsgrp  21344  slotsinbpsd  28419  slotslnbpsd  28420  setsvtx  29013  limsucncmpi  36487  sn-1ne2  42306