Theorem nnedc 2250

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 2246	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2	1	a1i 9	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵))
3	2	con2biidc 806	. 2 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴 ≠ 𝐵))
4	3	bicomd 139	1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 775   = wceq 1284   ≠ wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776  df-ne 2246

This theorem is referenced by:  nn0n0n1ge2b  8427  alzdvds  10254  fzo0dvdseq  10257  algcvgblem  10431