Theorem neeqtrri 2386

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 2191	. 2 |- B = C
4	1, 3	neeqtri 2384	1 |- A =/= C

Syntax hints:   = wceq 1363   =/= wne 2357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-cleq 2180  df-ne 2358

This theorem is referenced by:  pnfnemnf  8025  basendxnplusgndx  12597  plusgndxnmulrndx  12605  basendxnmulrndx  12606