Metamath Proof Explorer
Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003) Avoid ax-11 , ax-12 . (Revised by SN, 14-May-2025)
|
|
Ref |
Expression |
|
Assertion |
nfiu1 |
⊢ Ⅎ 𝑥 ∪ 𝑥 ∈ 𝐴 𝐵 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
eliun |
⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
2 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
3 |
1 2
|
nfxfr |
⊢ Ⅎ 𝑥 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 |
4 |
3
|
nfci |
⊢ Ⅎ 𝑥 ∪ 𝑥 ∈ 𝐴 𝐵 |