Theorem eqnetrri 2536

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 2357	. 2 ⊢ B = A
3		eqnetrr.2	. 2 ⊢ A ≠ C
4	2, 3	eqnetri 2534	1 ⊢ B ≠ C

This proof depends on syntax axioms:   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is used by:  ce0addcnnul  6180  addceq0  6220  1ne0c  6242  2ne0c  6243