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Mirrors > Home > MPE Home > Th. List > reuv | Structured version Visualization version GIF version |
Description: A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
Ref | Expression |
---|---|
reuv | ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3112 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3440 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 531 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | eubii 2630 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 279 | 1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2081 ∃!weu 2611 ∃!wreu 3107 Vcvv 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-reu 3112 df-v 3439 |
This theorem is referenced by: euen1 8427 updjud 9209 hlimeui 28708 |
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