Theorem nnedc 2425

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 2421	. . . 4 |- ( A =/= B <-> -. A = B )
2	1	a1i 9	. . 3 |- (DECID A = B -> ( A =/= B <-> -. A = B ) )
3	2	con2biidc 891	. 2 |- (DECID A = B -> ( A = B <-> -. A =/= B ) )
4	3	bicomd 141	1 |- (DECID A = B -> ( -. A =/= B <-> A = B ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 846   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-dc 847  df-ne 2421

This theorem is used by:  2omotaplemap  7623  nn0n0n1ge2b  9725  alzdvds  12621  fzo0dvdseq  12624  algcvgblem  12827