Theorem necon4abid 2981

Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon4abid.1	⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))

Assertion

Ref	Expression
necon4abid	⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))

Proof of Theorem necon4abid

Step	Hyp	Ref	Expression
1		notnotb 315	. 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
2		necon4abid.1	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))
3	2	necon1bbid 2980	. 2 ⊢ (𝜑 → (¬ ¬ 𝜓 ↔ 𝐴 = 𝐵))
4	1, 3	bitr2id 284	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ≠ wne 2940

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 2941

This theorem is referenced by:  necon4bbid  2982  necon2bbid  2984  birthdaylem3  26996  lgsprme0  27383  nmounbi  30795  disjecxrn  38390