Theorem eqnetri 2363

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 2355	. 2 |- ( A =/= C <-> B =/= C )
4	1, 3	mpbir 145	1 |- A =/= C

Syntax hints:   = wceq 1348   =/= wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by:  eqnetrri  2365  2on0  6405  1n0  6411  basendxnplusgndx  12524  plusgndxnmulrndx  12531  basendxnmulrndx  12532