Theorem necon3i 2555

Description: Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Hypothesis

Ref	Expression
necon3i.1	⊢ (A = B → C = D)

Assertion

Ref	Expression
necon3i	⊢ (C ≠ D → A ≠ B)

Proof of Theorem necon3i

Step	Hyp	Ref	Expression
1		necon3i.1	. 2 ⊢ (A = B → C = D)
2		id 19	. . 3 ⊢ ((A = B → C = D) → (A = B → C = D))
3	2	necon3d 2554	. 2 ⊢ ((A = B → C = D) → (C ≠ D → A ≠ B))
4	1, 3	ax-mp 8	1 ⊢ (C ≠ D → A ≠ B)

Syntax hints:   → wi 4   = wceq 1642   ≠ wne 2516

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

This theorem depends on definitions:  df-bi 177  df-ne 2518

This theorem is referenced by:  addcnnul  4453  tfin11  4493  eventfin  4517  oddtfin  4518  xpnz  5045