Theorem nnedc 2272

Description: Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)

Assertion

Ref	Expression
nnedc	⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem nnedc

Step	Hyp	Ref	Expression
1		df-ne 2268	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2	1	a1i 9	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵))
3	2	con2biidc 817	. 2 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴 ≠ 𝐵))
4	3	bicomd 140	1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 786   = wceq 1299   ≠ wne 2267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671

This theorem depends on definitions:  df-bi 116  df-dc 787  df-ne 2268

This theorem is referenced by:  nn0n0n1ge2b  8982  alzdvds  11347  fzo0dvdseq  11350  algcvgblem  11523