Theorem necon1i 2561

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Hypothesis

Ref	Expression
necon1i.1	⊢ (A ≠ B → C = D)

Assertion

Ref	Expression
necon1i	⊢ (C ≠ D → A = B)

Proof of Theorem necon1i

Step	Hyp	Ref	Expression
1		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1i.1	. . 3 ⊢ (A ≠ B → C = D)
3	1, 2	sylbir 204	. 2 ⊢ (¬ A = B → C = D)
4	3	necon1ai 2559	1 ⊢ (C ≠ D → A = B)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

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

This theorem depends on definitions:  df-bi 177  df-ne 2519

This theorem is referenced by:  map0b  6025