Theorem dcne 2256

Description: Decidable equality expressed in terms of =/=. Basically the same as df-dc 776. (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 776	. 2 |- (DECID A = B <-> ( A = B \/ -. A = B ) )
2		df-ne 2246	. . 3 |- ( A =/= B <-> -. A = B )
3	2	orbi2i 711	. 2 |- ( ( A = B \/ A =/= B ) <-> ( A = B \/ -. A = B ) )
4	1, 3	bitr4i 185	1 |- (DECID A = B <-> ( A = B \/ A =/= B ) )

Syntax hints:   -. wn 3   <-> wb 103   \/ wo 661  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-io 662

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

This theorem is referenced by:  zdceq  8423  nn0lt2  8429  qdceq  9256  nn0seqcvgd  10423