Theorem nnedc 2250

Description: Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)

Assertion

Ref	Expression
nnedc	|- (DECID A = B -> ( -. A =/= B <-> A = B ) )

Proof of Theorem nnedc

Step	Hyp	Ref	Expression
1		df-ne 2246	. . . 4 |- ( A =/= B <-> -. A = B )
2	1	a1i 9	. . 3 |- (DECID A = B -> ( A =/= B <-> -. A = B ) )
3	2	con2biidc 806	. 2 |- (DECID A = B -> ( A = B <-> -. A =/= B ) )
4	3	bicomd 139	1 |- (DECID A = B -> ( -. A =/= B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103  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-in1 576  ax-in2 577  ax-io 662

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

This theorem is referenced by:  nn0n0n1ge2b  8427  alzdvds  10254  fzo0dvdseq  10257  algcvgblem  10431