Theorem necon2abii 3011

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 3009	. 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 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:  frxp2  8149  frxp3  8156  locfindis  23724  flimsncls  24180  tsmsgsum  24333  wilthlem2  27270  karddom  35598  kardsdom  35599  topdifinffinlem  38034  ismblfin  38353  elnev  45188