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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 3379 | . 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) | |
| 2 | vex 3483 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) | 
| 4 | 3 | mobii 2547 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) | 
| 5 | 1, 4 | bitr4i 278 | 1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∃*wmo 2537 ∃*wrmo 3378 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-rmo 3379 df-v 3481 | 
| This theorem is referenced by: (None) | 
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