Theorem necon3bbii 2404

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 132	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝜑)
3	2	necon3abii 2403	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)
4	3	bicomi 132	1 ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 105   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2368

This theorem is referenced by:  ef0lem  11825