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Mirrors > Home > MPE Home > Th. List > rmov | Structured version Visualization version GIF version |
Description: An at-most-one 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 3146 | . 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 | mobii 2627 | . 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 2110 ∃*wmo 2616 ∃*wrmo 3141 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-rmo 3146 df-v 3496 |
This theorem is referenced by: (None) |
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