Theorem neeqtri 2367

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 2356	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)
4	1, 3	mpbi 144	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by:  neeqtrri  2369