Theorem dcbii 785

Description: The equivalent of a decidable proposition is decidable. (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	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	notbii 629	. . 3 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
3	1, 2	orbi12i 716	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓))
4		df-dc 781	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5		df-dc 781	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
6	3, 4, 5	3bitr4i 210	1 ⊢ (DECID 𝜑 ↔ DECID 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 103   ∨ wo 664  DECID wdc 780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781

This theorem is referenced by:  dcbi  882  dcned  2261  dfrex2dc  2371  euxfr2dc  2800  exmidexmid  4031  exfzdc  9647  nninfsellemdc  11857