Theorem neeqtri 2538

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

Hypotheses

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

Assertion

Ref	Expression
neeqtri	⊢ A ≠ C

Proof of Theorem neeqtri

Step	Hyp	Ref	Expression
1		neeqtr.1	. 2 ⊢ A ≠ B
2		neeqtr.2	. . 3 ⊢ B = C
3	2	neeq2i 2528	. 2 ⊢ (A ≠ B ↔ A ≠ C)
4	1, 3	mpbi 199	1 ⊢ A ≠ C

Syntax hints:   = wceq 1642   ≠ wne 2517

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is referenced by:  neeqtrri  2540