Theorem necon2bbid 2990

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 2987	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ≠ wne 2946

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 2947

This theorem is referenced by:  necon4bid  2992  fvdifsupp  8212  omwordi  8627  omass  8636  nnmwordi  8691  sdom1OLD  9306  pceq0  16918  f1otrspeq  19489  pmtrfinv  19503  symggen  19512  psgnunilem1  19535  mdetralt  22635  mdetunilem7  22645  ftalem5  27138  fsumvma  27275  dchrelbas4  27305  nosepssdm  27749  creq0  32749  fsumcvg4  33896  lkreqN  39126  flt4lem5elem  42606