Theorem eqnetri 2443

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 2435	. 2 |- ( A =/= C <-> B =/= C )
4	1, 3	mpbir 146	1 |- A =/= C

This proof depends on syntax axioms:   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is used by:  eqnetrri  2445  2on0  6697  1n0  6705  basendxnplusgndx  13479  plusgndxnmulrndx  13487  basendxnmulrndx  13488