Theorem dfrex2dc 2541

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 2534	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	dcbii 852	. . 3 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 ↔ DECID ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		dfexdc 1554	. . 3 ⊢ (DECID ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4	2, 3	sylbi 121	. 2 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)))
5		df-ral 2533	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
6		imnan 701	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	albii 1523	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
8	5, 7	bitri 184	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
9	8	notbii 678	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
10	4, 1, 9	3bitr4g 223	1 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846  ∀wal 1400  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-gen 1502  ax-ie2 1547

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-ral 2533  df-rex 2534

This theorem is referenced by:  dfrex2fin  7202  exmidomniim  7475