Theorem necon3abii 2372

Description: Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Hypothesis

Ref	Expression
necon3abii.1	⊢ (𝐴 = 𝐵 ↔ 𝜑)

Assertion

Ref	Expression
necon3abii	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)

Proof of Theorem necon3abii

Step	Hyp	Ref	Expression
1		df-ne 2337	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3abii.1	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝜑)
3	1, 2	xchbinx 672	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)

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:  necon3bbii  2373  necon3bii  2374  nesym  2381  n0rf  3421  gcd0id  11912