Theorem dcbii 845

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 844	. 2 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (DECID 𝜑 ↔ DECID 𝜓)

Syntax hints:   ↔ wb 105  DECID wdc 839

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840

This theorem is referenced by:  dcbi  942  dcned  2406  dfrex2dc  2521  euxfr2dc  2988  exmidexmid  4279  pw1fin  7068  tpfidceq  7088  dcfi  7144  elnn0dc  9802  elnndc  9803  exfzdc  10441  fprod1p  12105  bitsinv1  12468  nnwosdc  12555  prmdc  12647  pclemdc  12806  4sqlemafi  12913  4sqleminfi  12915  4sqexercise1  12916  nninfdclemcl  13014  nninfdclemp1  13016  psr1clfi  14646  nninfsellemdc  16335