Step |
Hyp |
Ref |
Expression |
1 |
|
axrep4v |
⊢ ( ∀ 𝑠 ∃ 𝑧 ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) → ∃ 𝑧 ∀ 𝑤 ( 𝑤 ∈ 𝑧 ↔ ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) ) |
2 |
|
ifptru |
⊢ ( ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ↔ 𝑤 = 𝑥 ) ) |
3 |
2
|
biimpd |
⊢ ( ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑥 ) ) |
4 |
|
equeuclr |
⊢ ( 𝑧 = 𝑥 → ( 𝑤 = 𝑥 → 𝑤 = 𝑧 ) ) |
5 |
3 4
|
syl9r |
⊢ ( 𝑧 = 𝑥 → ( ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) ) |
6 |
5
|
alrimdv |
⊢ ( 𝑧 = 𝑥 → ( ∃ 𝑛 𝑛 ∈ 𝑠 → ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) ) |
7 |
6
|
spimevw |
⊢ ( ∃ 𝑛 𝑛 ∈ 𝑠 → ∃ 𝑧 ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) |
8 |
|
ifpfal |
⊢ ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ↔ 𝑤 = 𝑦 ) ) |
9 |
8
|
biimpd |
⊢ ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑦 ) ) |
10 |
|
equeuclr |
⊢ ( 𝑧 = 𝑦 → ( 𝑤 = 𝑦 → 𝑤 = 𝑧 ) ) |
11 |
9 10
|
syl9r |
⊢ ( 𝑧 = 𝑦 → ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) ) |
12 |
11
|
alrimdv |
⊢ ( 𝑧 = 𝑦 → ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 → ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) ) |
13 |
12
|
spimevw |
⊢ ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 → ∃ 𝑧 ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) |
14 |
7 13
|
pm2.61i |
⊢ ∃ 𝑧 ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) |
15 |
1 14
|
mpg |
⊢ ∃ 𝑧 ∀ 𝑤 ( 𝑤 ∈ 𝑧 ↔ ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) |