Theorem necon2abii 2832

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)

Hypothesis

Ref	Expression
necon2abii.1	⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)

Assertion

Ref	Expression
necon2abii	⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)

Proof of Theorem necon2abii

Step	Hyp	Ref	Expression
1		necon2abii.1	. . . 4 ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)
2	1	bicomi 213	. . 3 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
3	2	necon1abii 2830	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
4	3	bicomi 213	1 ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ≠ wne 2780

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 196  df-ne 2782

This theorem is referenced by:  locfindis  21143  flimsncls  21600  tsmsgsum  21752  wilthlem2  24595  topdifinffinlem  32371  ismblfin  32620  elnev  37661