Theorem necon1abid 2994

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 318	. 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
2		necon1abid.1	. . 3 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))
3	2	necon3bbid 2993	. 2 ⊢ (𝜑 → (¬ ¬ 𝜓 ↔ 𝐴 ≠ 𝐵))
4	1, 3	bitr2id 287	1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ≠ wne 2956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ne 2957

This theorem is used by:  sotrine  5599  lttri2  11373  xrlttri2  13252  ioon0  13483  lssne0  21206  xmetgt0  24657