Theorem necon1bbii 3033

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 2988	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
2		necon1bbii.1	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
3	1, 2	xchnxbi 333	1 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1522   ≠ wne 2984

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 208  df-ne 2985

This theorem is referenced by:  necon2bbii  3035  intnex  5132  class2set  5145  csbopab  5330  relimasn  5828  modom  8565  supval2  8765  fzo0  12911  vma1  25425  lgsquadlem3  25640  ordtconnlem1  30784