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  2373  dfrex2dc  2488  euxfr2dc  2949  exmidexmid  4230  pw1fin  6980  tpfidceq  7000  dcfi  7056  elnn0dc  9704  elnndc  9705  exfzdc  10335  fprod1p  11783  bitsinv1  12146  nnwosdc  12233  prmdc  12325  pclemdc  12484  4sqlemafi  12591  4sqleminfi  12593  4sqexercise1  12594  nninfdclemcl  12692  nninfdclemp1  12694  psr1clfi  14322  nninfsellemdc  15765