Theorem eqnetri 2534

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 2527	. 2 ⊢ (A ≠ C ↔ B ≠ C)
4	1, 3	mpbir 200	1 ⊢ A ≠ C

Syntax hints:   = wceq 1642   ≠ wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is referenced by:  eqnetrri  2536  tfin1c  4500  map0  6026  ce0nnulb  6183  addceq0  6220  nchoicelem14  6303