Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1185.1 |
⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ) |
2 |
|
breq1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 𝑅 𝑧 ↔ 𝑦 𝑅 𝑧 ) ) |
3 |
2
|
notbid |
⊢ ( 𝑤 = 𝑦 → ( ¬ 𝑤 𝑅 𝑧 ↔ ¬ 𝑦 𝑅 𝑧 ) ) |
4 |
3
|
cbvralvw |
⊢ ( ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) |
5 |
4
|
rexbii |
⊢ ( ∃ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ↔ ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) |
6 |
1 5
|
sylib |
⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) |
7 |
|
eleq1w |
⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) ) |
8 |
|
breq2 |
⊢ ( 𝑧 = 𝑥 → ( 𝑦 𝑅 𝑧 ↔ 𝑦 𝑅 𝑥 ) ) |
9 |
8
|
notbid |
⊢ ( 𝑧 = 𝑥 → ( ¬ 𝑦 𝑅 𝑧 ↔ ¬ 𝑦 𝑅 𝑥 ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑧 = 𝑥 → ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) |
11 |
7 10
|
anbi12d |
⊢ ( 𝑧 = 𝑥 → ( ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) ) |
12 |
11
|
cbvexvw |
⊢ ( ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) |
13 |
|
df-rex |
⊢ ( ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) ) |
14 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) |
15 |
12 13 14
|
3bitr4ri |
⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) |
16 |
6 15
|
sylibr |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) |