Theorem necon2bbid 2975

Description: Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon2bbid.1	⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))

Assertion

Ref	Expression
necon2bbid	⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Proof of Theorem necon2bbid

Step	Hyp	Ref	Expression
1		necon2bbid.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))
2		notnotb 315	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3	1, 2	bitr3di 286	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓))
4	3	necon4abid 2972	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ≠ wne 2932

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 207  df-ne 2933

This theorem is referenced by:  necon4bid  2977  fvdifsupp  8121  omwordi  8506  omass  8515  nnmwordi  8571  pceq0  16842  f1otrspeq  19422  pmtrfinv  19436  symggen  19445  psgnunilem1  19468  mdetralt  22573  mdetunilem7  22583  ftalem5  27040  fsumvma  27176  dchrelbas4  27206  nosepssdm  27650  creq0  32809  suppgsumssiun  33133  fsumcvg4  34094  lkreqN  39616  flt4lem5elem  43084