Theorem dcbid 781

Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)

Hypothesis

Ref	Expression
dcbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
dcbid	⊢ (𝜑 → (DECID 𝜓 ↔ DECID 𝜒))

Proof of Theorem dcbid

Step	Hyp	Ref	Expression
1		dcbid.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	notbid 624	. . 3 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
3	1, 2	orbi12d 739	. 2 ⊢ (𝜑 → ((𝜓 ∨ ¬ 𝜓) ↔ (𝜒 ∨ ¬ 𝜒)))
4		df-dc 776	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
5		df-dc 776	. 2 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
6	3, 4, 5	3bitr4g 221	1 ⊢ (𝜑 → (DECID 𝜓 ↔ DECID 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   ∨ wo 661  DECID wdc 775

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 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  ltdcpi  6513  enqdc  6551  enqdc1  6552  ltdcnq  6587  exfzdc  9249  dvdsdc  10203  zdvdsdc  10216  zsupcllemstep  10341  infssuzex  10345