Theorem necon2abii 3007

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 227	. . 3 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
3	2	necon1abii 3005	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
4	3	bicomi 227	1 ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ≠ wne 2957

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 2958

This theorem is used by:  frxp2  8146  frxp3  8153  locfindis  23762  flimsncls  24218  tsmsgsum  24371  wilthlem2  27313  karddom  35695  kardsdom  35696  topdifinffinlem  38109  ismblfin  38418  elnev  45269