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Metamath Proof Explorer
Description: An existential quantifier restricted to the universe is unrestricted.
(Contributed by NM, 26-Mar-2004)
|
|
Ref |
Expression |
|
Assertion |
rexv |
⊢ ( ∃ 𝑥 ∈ V 𝜑 ↔ ∃ 𝑥 𝜑 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ V 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ V ∧ 𝜑 ) ) |
| 2 |
|
vex |
⊢ 𝑥 ∈ V |
| 3 |
2
|
biantrur |
⊢ ( 𝜑 ↔ ( 𝑥 ∈ V ∧ 𝜑 ) ) |
| 4 |
3
|
exbii |
⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ V ∧ 𝜑 ) ) |
| 5 |
1 4
|
bitr4i |
⊢ ( ∃ 𝑥 ∈ V 𝜑 ↔ ∃ 𝑥 𝜑 ) |