Theorem bj-dcbi 10719

Description: Equivalence property for DECID. TODO: solve conflict with dcbi 877; minimize dcbii 780 and dcbid 781 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dcbi	⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))

Proof of Theorem bj-dcbi

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2		bj-notbi 10716	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	orbi12d 739	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓)))
4		df-dc 776	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5		df-dc 776	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
6	3, 4, 5	3bitr4g 221	1 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))

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:  bj-d0clsepcl  10720