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Mirrors > Home > MPE Home > Th. List > reuv | Structured version Visualization version GIF version |
Description: A uniqueness 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 3057 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3343 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 528 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | eubii 2629 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 267 | 1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∈ wcel 2139 ∃!weu 2607 ∃!wreu 3052 Vcvv 3340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-12 2196 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-clab 2747 df-cleq 2753 df-clel 2756 df-reu 3057 df-v 3342 |
This theorem is referenced by: euen1 8193 updjud 8970 hlimeui 28427 |
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