Theorem dcne 2320

Description: Decidable equality expressed in terms of ≠. Basically the same as df-dc 821. (Contributed by Jim Kingdon, 14-Mar-2020.)

Assertion

Ref	Expression
dcne	⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))

Proof of Theorem dcne

Step	Hyp	Ref	Expression
1		df-dc 821	. 2 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
2		df-ne 2310	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	orbi2i 752	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
4	1, 3	bitr4i 186	1 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 698  DECID wdc 820   = wceq 1332   ≠ wne 2309

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-io 699

This theorem depends on definitions:  df-bi 116  df-dc 821  df-ne 2310

This theorem is referenced by:  updjudhf  6972  zdceq  9150  nn0lt2  9156  xlesubadd  9696  qdceq  10055  xrmaxadd  11062  nn0seqcvgd  11758