Theorem necon2bbid 2982

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 2979	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ≠ wne 2938

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 2939

This theorem is referenced by:  necon4bid  2984  fvdifsupp  8195  omwordi  8608  omass  8617  nnmwordi  8672  sdom1OLD  9277  pceq0  16905  f1otrspeq  19480  pmtrfinv  19494  symggen  19503  psgnunilem1  19526  mdetralt  22630  mdetunilem7  22640  ftalem5  27135  fsumvma  27272  dchrelbas4  27302  nosepssdm  27746  creq0  32753  fsumcvg4  33911  lkreqN  39152  flt4lem5elem  42638