Theorem necon2bbid 2968

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 2965	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ≠ wne 2925

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 2926

This theorem is referenced by:  necon4bid  2970  fvdifsupp  8111  omwordi  8496  omass  8505  nnmwordi  8560  pceq0  16801  f1otrspeq  19344  pmtrfinv  19358  symggen  19367  psgnunilem1  19390  mdetralt  22511  mdetunilem7  22521  ftalem5  27003  fsumvma  27140  dchrelbas4  27170  nosepssdm  27614  creq0  32692  fsumcvg4  33919  lkreqN  39151  flt4lem5elem  42627