Theorem necon1abii 2991

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Hypothesis

Ref	Expression
necon1abii.1	⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)

Assertion

Ref	Expression
necon1abii	⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)

Proof of Theorem necon1abii

Step	Hyp	Ref	Expression
1		notnotb 314	. 2 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2		necon1abii.1	. . 3 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
3	2	necon3bbii 2990	. 2 ⊢ (¬ ¬ 𝜑 ↔ 𝐴 ≠ 𝐵)
4	1, 3	bitr2i 275	1 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)

Syntax hints:  ¬ wn 3   ↔ 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:  necon2abii  2993  marypha1lem  9122  npomex  10683  uniinn0  30791