Theorem necon1bbii 3005

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 2960	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
2		necon1bbii.1	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
3	1, 2	xchnxbi 335	1 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)

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:  necon2bbii  3007  intnex  5306  class2set  5316  csbopab  5530  modom  9226  supval2  9431  fzo0  13798  vma1  27475  lgsquadlem3  27691  ordtconnlem1  34538