Theorem dcne 2273

Description: Decidable equality expressed in terms of =/=. Basically the same as df-dc 784. (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 784	. 2 |- (DECID A = B <-> ( A = B \/ -. A = B ) )
2		df-ne 2263	. . 3 |- ( A =/= B <-> -. A = B )
3	2	orbi2i 717	. 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 667  DECID wdc 783   = wceq 1296   =/= wne 2262

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668

This theorem depends on definitions:  df-bi 116  df-dc 784  df-ne 2263

This theorem is referenced by:  updjudhf  6850  zdceq  8920  nn0lt2  8926  xlesubadd  9449  qdceq  9807  xrmaxadd  10820  nn0seqcvgd  11465