Theorem neeqtri 3015

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 3008	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)
4	1, 3	mpbi 229	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1539   ≠ wne 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-ne 2943

This theorem is referenced by:  neeqtrri  3016  sn-0ne2  40310