Step |
Hyp |
Ref |
Expression |
1 |
|
3rspcedvd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |
2 |
|
3rspcedvd.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 ) |
3 |
|
3rspcedvd.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐷 ) |
4 |
|
3rspcedvd.1 |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
5 |
|
3rspcedvd.2 |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝜒 ↔ 𝜃 ) ) |
6 |
|
3rspcedvd.3 |
⊢ ( ( 𝜑 ∧ 𝑧 = 𝐶 ) → ( 𝜃 ↔ 𝜏 ) ) |
7 |
|
3rspcedvd.4 |
⊢ ( 𝜑 → 𝜏 ) |
8 |
4
|
2rexbidv |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜓 ↔ ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜒 ) ) |
9 |
5
|
rexbidv |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑧 ∈ 𝐷 𝜒 ↔ ∃ 𝑧 ∈ 𝐷 𝜃 ) ) |
10 |
3 6 7
|
rspcedvd |
⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝐷 𝜃 ) |
11 |
2 9 10
|
rspcedvd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜒 ) |
12 |
1 8 11
|
rspcedvd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐷 ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜓 ) |