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Mirrors > Home > MPE Home > Th. List > reueubd | Structured version Visualization version GIF version |
Description: Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.) |
Ref | Expression |
---|---|
reueubd.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉) |
Ref | Expression |
---|---|
reueubd | ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3113 | . 2 ⊢ (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓)) | |
2 | reueubd.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉) | |
3 | 2 | ex 416 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝑉)) |
4 | 3 | pm4.71rd 566 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝑥 ∈ 𝑉 ∧ 𝜓))) |
5 | 4 | eubidv 2647 | . 2 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝑉 ∧ 𝜓))) |
6 | 1, 5 | bitr4id 293 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∃!weu 2628 ∃!wreu 3108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-mo 2598 df-eu 2629 df-reu 3113 |
This theorem is referenced by: frgreu 28053 |
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