Step |
Hyp |
Ref |
Expression |
1 |
|
cbvral4vw.1 |
⊢ ( 𝑥 = 𝑎 → ( 𝜑 ↔ 𝜒 ) ) |
2 |
|
cbvral4vw.2 |
⊢ ( 𝑦 = 𝑏 → ( 𝜒 ↔ 𝜃 ) ) |
3 |
|
cbvral4vw.3 |
⊢ ( 𝑧 = 𝑐 → ( 𝜃 ↔ 𝜏 ) ) |
4 |
|
cbvral4vw.4 |
⊢ ( 𝑤 = 𝑑 → ( 𝜏 ↔ 𝜓 ) ) |
5 |
1
|
ralbidv |
⊢ ( 𝑥 = 𝑎 → ( ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑤 ∈ 𝐷 𝜒 ) ) |
6 |
2
|
ralbidv |
⊢ ( 𝑦 = 𝑏 → ( ∀ 𝑤 ∈ 𝐷 𝜒 ↔ ∀ 𝑤 ∈ 𝐷 𝜃 ) ) |
7 |
3
|
ralbidv |
⊢ ( 𝑧 = 𝑐 → ( ∀ 𝑤 ∈ 𝐷 𝜃 ↔ ∀ 𝑤 ∈ 𝐷 𝜏 ) ) |
8 |
5 6 7
|
cbvral3vw |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜏 ) |
9 |
4
|
cbvralvw |
⊢ ( ∀ 𝑤 ∈ 𝐷 𝜏 ↔ ∀ 𝑑 ∈ 𝐷 𝜓 ) |
10 |
9
|
3ralbii |
⊢ ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜏 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑑 ∈ 𝐷 𝜓 ) |
11 |
8 10
|
bitri |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑑 ∈ 𝐷 𝜓 ) |