Theorem eqnetrri 2230

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 2044	. 2 ⊢ 𝐵 = 𝐴
3		eqnetrr.2	. 2 ⊢ 𝐴 ≠ 𝐶
4	2, 3	eqnetri 2228	1 ⊢ 𝐵 ≠ 𝐶

Syntax hints:   = wceq 1243   ≠ wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206

This theorem is referenced by: (None)