Theorem bj-dcbi 12123

Description: Equivalence property for DECID. TODO: solve conflict with dcbi 883; minimize dcbii 786 and dcbid 787 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 12120	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	orbi12d 743	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓)))
4		df-dc 782	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5		df-dc 782	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
6	3, 4, 5	3bitr4g 222	1 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 665  DECID wdc 781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666

This theorem depends on definitions:  df-bi 116  df-dc 782

This theorem is referenced by:  bj-d0clsepcl  12124