Theorem necon2bbii 3012

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon2bbii.1	⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
necon2bbii	⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)

Proof of Theorem necon2bbii

Step	Hyp	Ref	Expression
1		necon2bbii.1	. . . 4 ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)
2	1	bicomi 227	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
3	2	necon1bbii 3010	. 2 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
4	3	bicomi 227	1 ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ≠ wne 2961

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 2962

This theorem is used by:  xpeq0  6162  dmsn0  6215  disjex  32974  disjexc  32975  suppss3  33105  map0cor  49674