Theorem eqnetrri 2392

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

Hypotheses

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

Assertion

Ref	Expression
eqnetrri	⊢ 𝐵 ≠ 𝐶

Proof of Theorem eqnetrri

Step	Hyp	Ref	Expression
1		eqnetrr.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	eqcomi 2200	. 2 ⊢ 𝐵 = 𝐴
3		eqnetrr.2	. 2 ⊢ 𝐴 ≠ 𝐶
4	2, 3	eqnetri 2390	1 ⊢ 𝐵 ≠ 𝐶

Syntax hints:   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by: (None)