![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3114 | . 2 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3444 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 534 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | mobii 2606 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 281 | 1 ⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∃*wmo 2596 ∃*wrmo 3109 Vcvv 3441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-mo 2598 df-clab 2777 df-cleq 2791 df-clel 2870 df-rmo 3114 df-v 3443 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |