Theorem dcbii 847

Description: Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.)

Hypothesis

Ref	Expression
dcbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
dcbii	⊢ (DECID 𝜑 ↔ DECID 𝜓)

Proof of Theorem dcbii

Step	Hyp	Ref	Expression
1		dcbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		dcbiit 846	. 2 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (DECID 𝜑 ↔ DECID 𝜓)

Syntax hints:   ↔ wb 105  DECID wdc 841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-dc 842

This theorem is referenced by:  dcbi  944  dcned  2408  dfrex2dc  2523  euxfr2dc  2991  exmidexmid  4286  pw1fin  7102  tpfidceq  7122  dcfi  7180  elnn0dc  9845  elnndc  9846  exfzdc  10487  fprod1p  12165  bitsinv1  12528  nnwosdc  12615  prmdc  12707  pclemdc  12866  4sqlemafi  12973  4sqleminfi  12975  4sqexercise1  12976  nninfdclemcl  13074  nninfdclemp1  13076  psr1clfi  14708  nninfsellemdc  16638