Theorem bj-dcbi 10048

Description: Equivalence property for DECID. TODO: solve conflict with dcbi 844; minimize dcbii 747 and dcbid 748 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 10045	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	orbi12d 707	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((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	3bitr4g 212	1 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  bj-d0clsepcl  10049