Theorem neeqtrri 2389

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 2193	. 2 |- B = C
4	1, 3	neeqtri 2387	1 |- A =/= C

Syntax hints:   = wceq 1364   =/= wne 2360

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-ne 2361

This theorem is referenced by:  pnfnemnf  8030  basendxnplusgndx  12602  plusgndxnmulrndx  12610  basendxnmulrndx  12611