Theorem dcne 2216

Description: Decidable equality expressed in terms of =/=. Basically the same as df-dc 743. (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 743	. 2 |- (DECID A = B <-> ( A = B \/ -. A = B ) )
2		df-ne 2206	. . 3 |- ( A =/= B <-> -. A = B )
3	2	orbi2i 679	. 2 |- ( ( A = B \/ A =/= B ) <-> ( A = B \/ -. A = B ) )
4	1, 3	bitr4i 176	1 |- (DECID A = B <-> ( A = B \/ A =/= B ) )

Syntax hints:   -. wn 3   <-> wb 98   \/ wo 629  DECID wdc 742   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206

This theorem is referenced by:  zdceq  8316  nn0lt2  8322  qdceq  9102  nn0seqcvgd  9880