Theorem necon1d 2964

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 2946	. . 3 ⊢ (¬ 𝐶 ≠ 𝐷 ↔ 𝐶 = 𝐷)
3	1, 2	syl6ibr 251	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝐶 ≠ 𝐷))
4	3	necon4ad 2961	1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ≠ wne 2942

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 206  df-ne 2943

This theorem is referenced by:  disji  5053  mul02lem2  11082  mhpmulcl  21249  xblss2ps  23462  xblss2  23463  lgsne0  26388  h1datomi  29844  eigorthi  30100  disjif  30818  lineintmo  34386  poimirlem6  35710  poimirlem7  35711  2llnmat  37465  2lnat  37725  tendospcanN  38964  dihmeetlem13N  39260  dochkrshp  39327  remul02  40309  remul01  40311  sn-0tie0  40342