Theorem eqnetrri 3090

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 2833	. 2 ⊢ 𝐵 = 𝐴
3		eqnetrr.2	. 2 ⊢ 𝐴 ≠ 𝐶
4	2, 3	eqnetri 3089	1 ⊢ 𝐵 ≠ 𝐶

Syntax hints:   = wceq 1536   ≠ wne 3019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-9 2123  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-cleq 2817  df-ne 3020

This theorem is referenced by:  ballotlemii  31765  bj-2upln1upl  34340  wallispilem4  42360