Theorem necon3bbii 2373

Description: Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon3bbii.1	⊢ (𝜑 ↔ 𝐴 = 𝐵)

Assertion

Ref	Expression
necon3bbii	⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵)

Proof of Theorem necon3bbii

Step	Hyp	Ref	Expression
1		necon3bbii.1	. . . 4 ⊢ (𝜑 ↔ 𝐴 = 𝐵)
2	1	bicomi 131	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝜑)
3	2	necon3abii 2372	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)
4	3	bicomi 131	1 ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 104   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  ef0lem  11601