Theorem dcbii 841

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 840	. 2 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (DECID 𝜑 ↔ DECID 𝜓)

Syntax hints:   ↔ wb 105  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by:  dcbi  938  dcned  2370  dfrex2dc  2485  euxfr2dc  2945  exmidexmid  4225  pw1fin  6966  dcfi  7040  elnn0dc  9676  elnndc  9677  exfzdc  10307  fprod1p  11742  nnwosdc  12176  prmdc  12268  pclemdc  12426  4sqlemafi  12533  4sqleminfi  12535  4sqexercise1  12536  nninfdclemcl  12605  nninfdclemp1  12607  nninfsellemdc  15500