Theorem necon1abii 3004

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 318	. 2 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2		necon1abii.1	. . 3 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
3	2	necon3bbii 3003	. 2 ⊢ (¬ ¬ 𝜑 ↔ 𝐴 ≠ 𝐵)
4	1, 3	bitr2i 279	1 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ 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:  necon2abii  3006  marypha1lem  9418  npomex  11074  uniinn0  33140