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Mirrors > Home > ILE Home > Th. List > dfrex2dc | GIF version |
Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.) |
Ref | Expression |
---|---|
dfrex2dc | ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2381 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | dcbii 791 | . . 3 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 ↔ DECID ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
3 | dfexdc 1445 | . . 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 2380 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
6 | imnan 665 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
7 | 6 | albii 1414 | . . . 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 635 | . 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)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 786 ∀wal 1297 ∃wex 1436 ∈ wcel 1448 ∀wral 2375 ∃wrex 2376 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-gen 1393 ax-ie2 1438 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-tru 1302 df-fal 1305 df-ral 2380 df-rex 2381 |
This theorem is referenced by: dfrex2fin 6726 exmidomniim 6925 |
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