Theorem eqnetri 2303

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

Hypotheses

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

Assertion

Ref	Expression
eqnetri	⊢ 𝐴 ≠ 𝐶

Proof of Theorem eqnetri

Step	Hyp	Ref	Expression
1		eqnetr.2	. 2 ⊢ 𝐵 ≠ 𝐶
2		eqnetr.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	neeq1i 2295	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
4	1, 3	mpbir 145	1 ⊢ 𝐴 ≠ 𝐶

Syntax hints:   = wceq 1312   ≠ wne 2280

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1404  ax-gen 1406  ax-4 1468  ax-17 1487  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-cleq 2106  df-ne 2281

This theorem is referenced by:  eqnetrri  2305  2on0  6275  1n0  6281  basendxnplusgndx  11902  plusgndxnmulrndx  11909  basendxnmulrndx  11910