Theorem eqnetrri 3035

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 2778	. 2 ⊢ 𝐵 = 𝐴
3		eqnetrr.2	. 2 ⊢ 𝐴 ≠ 𝐶
4	2, 3	eqnetri 3034	1 ⊢ 𝐵 ≠ 𝐶

Syntax hints:   = wceq 1567   ≠ wne 2964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-ne 2965

This theorem is referenced by:  ballotlemii  34838  bj-2upln1upl  37547  sn-0tie0  43114  wallispilem4  46673