Theorem eqnetri 2390

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
eqnetr.1	|- A = B
eqnetr.2	|- B =/= C

Assertion

Ref	Expression
eqnetri	|- A =/= C

Proof of Theorem eqnetri

Step	Hyp	Ref	Expression
1		eqnetr.2	. 2 |- B =/= C
2		eqnetr.1	. . 3 |- A = B
3	2	neeq1i 2382	. 2 |- ( A =/= C <-> B =/= C )
4	1, 3	mpbir 146	1 |- A =/= C

Syntax hints:   = wceq 1364   =/= wne 2367

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  eqnetrri  2392  2on0  6484  1n0  6490  basendxnplusgndx  12802  plusgndxnmulrndx  12810  basendxnmulrndx  12811