Theorem nnedc 2372

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 2368	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2	1	a1i 9	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵))
3	2	con2biidc 880	. 2 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴 ≠ 𝐵))
4	3	bicomd 141	1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 835   = 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  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836  df-ne 2368

This theorem is referenced by:  2omotaplemap  7340  nn0n0n1ge2b  9422  alzdvds  12036  fzo0dvdseq  12039  algcvgblem  12242