Step |
Hyp |
Ref |
Expression |
1 |
|
opnneir.1 |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
2 |
|
opnneilv.2 |
⊢ ( ( 𝜑 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝜓 → 𝜒 ) ) |
3 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) ) |
4 |
|
neii2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ) |
5 |
1 4
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ) |
6 |
5
|
r19.41dv |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 𝜓 ) → ∃ 𝑦 ∈ 𝐽 ( ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ∧ 𝜓 ) ) |
7 |
6
|
expl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) → ∃ 𝑦 ∈ 𝐽 ( ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ∧ 𝜓 ) ) ) |
8 |
|
anass |
⊢ ( ( ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ∧ 𝜓 ) ↔ ( 𝑆 ⊆ 𝑦 ∧ ( 𝑦 ⊆ 𝑥 ∧ 𝜓 ) ) ) |
9 |
2
|
expimpd |
⊢ ( 𝜑 → ( ( 𝑦 ⊆ 𝑥 ∧ 𝜓 ) → 𝜒 ) ) |
10 |
9
|
anim2d |
⊢ ( 𝜑 → ( ( 𝑆 ⊆ 𝑦 ∧ ( 𝑦 ⊆ 𝑥 ∧ 𝜓 ) ) → ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) ) |
11 |
8 10
|
syl5bi |
⊢ ( 𝜑 → ( ( ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ∧ 𝜓 ) → ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) ) |
12 |
11
|
reximdv |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐽 ( ( 𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥 ) ∧ 𝜓 ) → ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) ) |
13 |
7 12
|
syld |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) → ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) ) |
14 |
13
|
exlimdv |
⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ∧ 𝜓 ) → ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) ) |
15 |
3 14
|
syl5bi |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜓 → ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) ) |