Step |
Hyp |
Ref |
Expression |
1 |
|
ax13lem1 |
⊢ ( ¬ 𝑥 = 𝑦 → ( 𝑤 = 𝑦 → ∀ 𝑥 𝑤 = 𝑦 ) ) |
2 |
|
equeucl |
⊢ ( 𝑧 = 𝑦 → ( 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ) |
3 |
2
|
eximi |
⊢ ( ∃ 𝑥 𝑧 = 𝑦 → ∃ 𝑥 ( 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ) |
4 |
|
19.36v |
⊢ ( ∃ 𝑥 ( 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ↔ ( ∀ 𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ) |
5 |
3 4
|
sylib |
⊢ ( ∃ 𝑥 𝑧 = 𝑦 → ( ∀ 𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ) |
6 |
1 5
|
syl9 |
⊢ ( ¬ 𝑥 = 𝑦 → ( ∃ 𝑥 𝑧 = 𝑦 → ( 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ) ) |
7 |
6
|
alrimdv |
⊢ ( ¬ 𝑥 = 𝑦 → ( ∃ 𝑥 𝑧 = 𝑦 → ∀ 𝑤 ( 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ) ) |
8 |
|
equequ2 |
⊢ ( 𝑤 = 𝑦 → ( 𝑧 = 𝑤 ↔ 𝑧 = 𝑦 ) ) |
9 |
8
|
equsalvw |
⊢ ( ∀ 𝑤 ( 𝑤 = 𝑦 → 𝑧 = 𝑤 ) ↔ 𝑧 = 𝑦 ) |
10 |
7 9
|
syl6ib |
⊢ ( ¬ 𝑥 = 𝑦 → ( ∃ 𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦 ) ) |