Theorem necon1abid 2982

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon1abid.1	⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))

Assertion

Ref	Expression
necon1abid	⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))

Proof of Theorem necon1abid

Step	Hyp	Ref	Expression
1		notnotb 315	. 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
2		necon1abid.1	. . 3 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))
3	2	necon3bbid 2981	. 2 ⊢ (𝜑 → (¬ ¬ 𝜓 ↔ 𝐴 ≠ 𝐵))
4	1, 3	bitr2id 284	1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ≠ wne 2943

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 206  df-ne 2944

This theorem is referenced by:  lttri2  11057  xrlttri2  12876  ioon0  13105  lssne0  20212  xmetgt0  23511  sotrine  33734