Theorem neeqtrri 3029

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 2770	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	neeqtri 3028	1 ⊢ 𝐴 ≠ 𝐶

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-ne 2957

This theorem is used by:  nlim1  8497  nlim2  8498  1one2o  8655  cflim2  10341  pnfnemnf  11364  basendxnmulrndx  17467  plusgndxnmulrndx  17468  slotsbhcdif  17586  xrsnsgrp  21714  slotsinbpsd  28903  slotslnbpsd  28904  setsvtx  29613  kard0b  35827  limsucncmpi  37233  sn-1ne2  43330