Theorem dcned 2212

Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)

Hypothesis

Ref	Expression
dcned.eq	|- ( ph -> DECID A = B )

Assertion

Ref	Expression
dcned	|- ( ph -> DECID A =/= B )

Proof of Theorem dcned

Step	Hyp	Ref	Expression
1		dcned.eq	. . 3 |- ( ph -> DECID A = B )
2		dcn 746	. . 3 |- (DECID A = B -> DECID -. A = B )
3	1, 2	syl 14	. 2 |- ( ph -> DECID -. A = B )
4		df-ne 2206	. . 3 |- ( A =/= B <-> -. A = B )
5	4	dcbii 747	. 2 |- (DECID A =/= B <-> DECID -. A = B )
6	3, 5	sylibr 137	1 |- ( ph -> DECID A =/= B )

Syntax hints:   -. wn 3   -> wi 4  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-in1 544  ax-in2 545  ax-io 630

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

This theorem is referenced by:  nn0n0n1ge2b  8320  algcvgblem  9888