Theorem neeqtrri 3007

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 2748	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3006	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1547   ≠ wne 2934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-ne 2935

This theorem is referenced by:  nlim1  8414  nlim2  8415  1one2o  8572  cflim2  10176  pnfnemnf  11191  basendxnmulrndx  17250  plusgndxnmulrndx  17251  slotsbhcdif  17369  xrsnsgrp  21383  slotsinbpsd  28527  slotslnbpsd  28528  setsvtx  29122  limsucncmpi  36673  sn-1ne2  42748