Theorem dcned 2209

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

Hypothesis

Ref	Expression
dcned.eq	⊢ (φ → DECID A = B)

Assertion

Ref	Expression
dcned	⊢ (φ → DECID A ≠ B)

Proof of Theorem dcned

Step	Hyp	Ref	Expression
1		dcned.eq	. . 3 ⊢ (φ → DECID A = B)
2		dcn 745	. . 3 ⊢ (DECID A = B → DECID ¬ A = B)
3	1, 2	syl 14	. 2 ⊢ (φ → DECID ¬ A = B)
4		df-ne 2203	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
5	4	dcbii 746	. 2 ⊢ (DECID A ≠ B ↔ DECID ¬ A = B)
6	3, 5	sylibr 137	1 ⊢ (φ → DECID A ≠ B)

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 741   = wceq 1242   ≠ wne 2201

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742  df-ne 2203

This theorem is referenced by:  nn0n0n1ge2b  8096