Theorem dcne 2319

Description: Decidable equality expressed in terms of =/=. Basically the same as df-dc 820. (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 820	. 2 |- (DECID A = B <-> ( A = B \/ -. A = B ) )
2		df-ne 2309	. . 3 |- ( A =/= B <-> -. A = B )
3	2	orbi2i 751	. 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 697  DECID wdc 819   = wceq 1331   =/= wne 2308

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 698

This theorem depends on definitions:  df-bi 116  df-dc 820  df-ne 2309

This theorem is referenced by:  updjudhf  6964  zdceq  9126  nn0lt2  9132  xlesubadd  9666  qdceq  10024  xrmaxadd  11030  nn0seqcvgd  11722