Theorem neeqtri 2272

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 2261	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)
4	1, 3	mpbi 143	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1284   ≠ wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-ne 2246

This theorem is referenced by:  neeqtrri  2274