Metamath Proof Explorer
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012) (Proof shortened by AV, 7-Apr-2023)
|
|
Ref |
Expression |
|
Hypothesis |
ralsng.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
rexsng |
⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ralsng.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
nfv |
⊢ Ⅎ 𝑥 𝜓 |
3 |
2 1
|
rexsngf |
⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) ) |