Theorem necon1d 2586

Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1d.1	⊢ (φ → (A ≠ B → C = D))

Assertion

Ref	Expression
necon1d	⊢ (φ → (C ≠ D → A = B))

Proof of Theorem necon1d

Step	Hyp	Ref	Expression
1		necon1d.1	. . 3 ⊢ (φ → (A ≠ B → C = D))
2		nne 2521	. . 3 ⊢ (¬ C ≠ D ↔ C = D)
3	1, 2	syl6ibr 218	. 2 ⊢ (φ → (A ≠ B → ¬ C ≠ D))
4	3	necon4ad 2578	1 ⊢ (φ → (C ≠ D → A = B))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is used by: (None)