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Mirrors > Home > ILE Home > Th. List > rmov | GIF version |
Description: A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
rmov | ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 2314 | . 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 287 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | mobii 1937 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 176 | 1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∈ wcel 1393 ∃*wmo 1901 ∃*wrmo 2309 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-rmo 2314 df-v 2559 |
This theorem is referenced by: (None) |
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