Theorem necon1bbii 3067

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon1bbii.1	⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)

Assertion

Ref	Expression
necon1bbii	⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)

Proof of Theorem necon1bbii

Step	Hyp	Ref	Expression
1		nne 3022	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
2		necon1bbii.1	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
3	1, 2	xchnxbi 334	1 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1537   ≠ wne 3018

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 209  df-ne 3019

This theorem is referenced by:  necon2bbii  3069  intnex  5243  class2set  5256  csbopab  5444  relimasn  5954  modom  8721  supval2  8921  fzo0  13064  vma1  25745  lgsquadlem3  25960  ordtconnlem1  31169