Theorem dcbii 747

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 594	. . 3 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
3	1, 2	orbi12i 681	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓))
4		df-dc 743	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5		df-dc 743	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
6	3, 4, 5	3bitr4i 201	1 ⊢ (DECID 𝜑 ↔ DECID 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  dcbi  844  dcned  2212  euxfr2dc  2726