Theorem neeqtrd 2539

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

Hypotheses

Ref	Expression
neeqtrd.1	⊢ (φ → A ≠ B)
neeqtrd.2	⊢ (φ → B = C)

Assertion

Ref	Expression
neeqtrd	⊢ (φ → A ≠ C)

Proof of Theorem neeqtrd

Step	Hyp	Ref	Expression
1		neeqtrd.1	. 2 ⊢ (φ → A ≠ B)
2		neeqtrd.2	. . 3 ⊢ (φ → B = C)
3	2	neeq2d 2531	. 2 ⊢ (φ → (A ≠ B ↔ A ≠ C))
4	1, 3	mpbid 201	1 ⊢ (φ → A ≠ C)

Syntax hints:   → wi 4   = 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:  neeqtrrd  2541