![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rexv | GIF version |
Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
rexv | ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2423 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 2692 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 301 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | exbii 1585 | . 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | bitr4i 186 | 1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1469 ∈ wcel 1481 ∃wrex 2418 Vcvv 2689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-rex 2423 df-v 2691 |
This theorem is referenced by: rexcom4 2712 spesbc 2998 abnex 4376 dfco2 5046 dfco2a 5047 |
Copyright terms: Public domain | W3C validator |