Theorem necon2bbid 2999

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 318	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3	1, 2	bitr3di 289	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓))
4	3	necon4abid 2996	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ≠ wne 2956

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

This proof depends on definitions:  df-bi 210  df-ne 2957

This theorem is used by:  necon4bid  3001  fvdifsupp  8181  omwordi  8572  omass  8581  nnmwordi  8637  pceq0  17042  f1otrspeq  19654  pmtrfinv  19668  symggen  19677  psgnunilem1  19700  mdetralt  22916  mdetunilem7  22926  ftalem5  27397  fsumvma  27533  dchrelbas4  27563  flt4lem5elem  27974  nosepssdm  28036  creq0  33321  suppgsumssiun  33626  fsumcvg4  34575  lkreqN  40207