Theorem nnedc 2345

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 2341	. . . 4 |- ( A =/= B <-> -. A = B )
2	1	a1i 9	. . 3 |- (DECID A = B -> ( A =/= B <-> -. A = B ) )
3	2	con2biidc 874	. 2 |- (DECID A = B -> ( A = B <-> -. A =/= B ) )
4	3	bicomd 140	1 |- (DECID A = B -> ( -. A =/= B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 829   = wceq 1348   =/= wne 2340

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-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830  df-ne 2341

This theorem is referenced by:  nn0n0n1ge2b  9291  alzdvds  11814  fzo0dvdseq  11817  algcvgblem  12003