Theorem necon1i 1613

Description: Contrapositive inference for inequality.

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 1590	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1i.1	. . 3 ⊢ (A ≠ B → C = D)
3	1, 2	sylbir 201	. 2 ⊢ (¬ A = B → C = D)
4	3	necon1ai 1611	1 ⊢ (C ≠ D → A = B)

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

This theorem is referenced by:  map0b 4349

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