Theorem dfrex2dc 2457

Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)

Assertion

Ref	Expression
dfrex2dc	⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑))

Proof of Theorem dfrex2dc

Step	Hyp	Ref	Expression
1		df-rex 2450	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	dcbii 830	. . 3 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 ↔ DECID ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		dfexdc 1489	. . 3 ⊢ (DECID ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4	2, 3	sylbi 120	. 2 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
5		df-ral 2449	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
6		imnan 680	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	albii 1458	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
8	5, 7	bitri 183	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
9	8	notbii 658	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
10	4, 1, 9	3bitr4g 222	1 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824  ∀wal 1341  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-gen 1437  ax-ie2 1482

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-ral 2449  df-rex 2450

This theorem is referenced by:  dfrex2fin  6869  exmidomniim  7105