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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 3145 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 533 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | eubii 2670 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 280 | 1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∃!weu 2653 ∃!wreu 3140 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-reu 3145 df-v 3496 |
This theorem is referenced by: euen1 8579 updjud 9363 hlimeui 29017 |
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