Theorem necon4bbid 2984

Description: Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)

Hypothesis

Ref	Expression
necon4bbid.1	⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))

Assertion

Ref	Expression
necon4bbid	⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))

Proof of Theorem necon4bbid

Step	Hyp	Ref	Expression
1		necon4bbid.1	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))
2	1	bicomd 222	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))
3	2	necon4abid 2983	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))
4	3	bicomd 222	1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ≠ wne 2942

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 2943

This theorem is referenced by:  fzn  13201  lgsqr  26404