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  4280  pw1fin  7083  tpfidceq  7103  dcfi  7159  elnn0dc  9818  elnndc  9819  exfzdc  10458  fprod1p  12125  bitsinv1  12488  nnwosdc  12575  prmdc  12667  pclemdc  12826  4sqlemafi  12933  4sqleminfi  12935  4sqexercise1  12936  nninfdclemcl  13034  nninfdclemp1  13036  psr1clfi  14667  nninfsellemdc  16436