Theorem neeqtri 3030

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 3023	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)
4	1, 3	mpbi 233	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1570   ≠ wne 2958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-ne 2959

This theorem is referenced by:  neeqtrri  3031  sn-0ne2  43145