Theorem necon4i 1628

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon4i.1	⊢ (A ≠ B → C ≠ D)

Assertion

Ref	Expression
necon4i	⊢ (C = D → A = B)

Proof of Theorem necon4i

Step	Hyp	Ref	Expression
1		necon4i.1	. . 3 ⊢ (A ≠ B → C ≠ D)
2		df-ne 1590	. . 3 ⊢ (C ≠ D ↔ ¬ C = D)
3	1, 2	sylib 198	. 2 ⊢ (A ≠ B → ¬ C = D)
4	3	necon4ai 1627	1 ⊢ (C = D → A = B)

Syntax hints:  ¬ wn 2   → wi 3   = wceq 958   ≠ wne 1588

This theorem is referenced by:  map0 4350  scott0 4727

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-ne 1590

Copyright terms: Public domain