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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 3143 | . 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3495 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 531 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | mobii 2624 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 279 | 1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∃*wmo 2613 ∃*wrmo 3138 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-mo 2615 df-clab 2797 df-cleq 2811 df-clel 2890 df-rmo 3143 df-v 3494 |
This theorem is referenced by: (None) |
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