Theorem necon1d 2803

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 2785	. . 3 ⊢ (¬ 𝐶 ≠ 𝐷 ↔ 𝐶 = 𝐷)
3	1, 2	syl6ibr 240	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝐶 ≠ 𝐷))
4	3	necon4ad 2800	1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1474   ≠ wne 2779

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 195  df-ne 2781

This theorem is referenced by:  disji  4564  mul02lem2  10064  xblss2ps  21957  xblss2  21958  lgsne0  24777  h1datomi  27630  eigorthi  27886  disjif  28579  lineintmo  31240  poimirlem6  32381  poimirlem7  32382  2llnmat  33624  2lnat  33884  tendospcanN  35126  dihmeetlem13N  35422  dochkrshp  35489