Theorem eqnetrri 2445

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

Hypotheses

Ref	Expression
eqnetrr.1	⊢ 𝐴 = 𝐵
eqnetrr.2	⊢ 𝐴 ≠ 𝐶

Assertion

Ref	Expression
eqnetrri	⊢ 𝐵 ≠ 𝐶

Proof of Theorem eqnetrri

Step	Hyp	Ref	Expression
1		eqnetrr.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	eqcomi 2242	. 2 ⊢ 𝐵 = 𝐴
3		eqnetrr.2	. 2 ⊢ 𝐴 ≠ 𝐶
4	2, 3	eqnetri 2443	1 ⊢ 𝐵 ≠ 𝐶

Syntax hints:   = wceq 1402   ≠ wne 2420

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

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

This theorem is referenced by:  ballotfilemii  13224