Theorem neeqtrri 3006

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 2746	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3005	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1542   ≠ wne 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ne 2934

This theorem is referenced by:  nlim1  8418  nlim2  8419  1one2o  8576  cflim2  10179  pnfnemnf  11194  basendxnmulrndx  17253  plusgndxnmulrndx  17254  slotsbhcdif  17372  xrsnsgrp  21400  slotsinbpsd  28526  slotslnbpsd  28527  setsvtx  29121  limsucncmpi  36646  sn-1ne2  42720