Metamath Proof Explorer
Description: 3-variable restricted specialization, using implicit substitution.
(Contributed by Scott Fenton, 10-Mar-2025)
|
|
Ref |
Expression |
|
Hypotheses |
rspc3dv.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) ) |
|
|
rspc3dv.2 |
⊢ ( 𝑦 = 𝐵 → ( 𝜃 ↔ 𝜏 ) ) |
|
|
rspc3dv.3 |
⊢ ( 𝑧 = 𝐶 → ( 𝜏 ↔ 𝜒 ) ) |
|
|
rspc3dv.4 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐸 ∀ 𝑧 ∈ 𝐹 𝜓 ) |
|
|
rspc3dv.5 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |
|
|
rspc3dv.6 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐸 ) |
|
|
rspc3dv.7 |
⊢ ( 𝜑 → 𝐶 ∈ 𝐹 ) |
|
Assertion |
rspc3dv |
⊢ ( 𝜑 → 𝜒 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rspc3dv.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) ) |
2 |
|
rspc3dv.2 |
⊢ ( 𝑦 = 𝐵 → ( 𝜃 ↔ 𝜏 ) ) |
3 |
|
rspc3dv.3 |
⊢ ( 𝑧 = 𝐶 → ( 𝜏 ↔ 𝜒 ) ) |
4 |
|
rspc3dv.4 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐸 ∀ 𝑧 ∈ 𝐹 𝜓 ) |
5 |
|
rspc3dv.5 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |
6 |
|
rspc3dv.6 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐸 ) |
7 |
|
rspc3dv.7 |
⊢ ( 𝜑 → 𝐶 ∈ 𝐹 ) |
8 |
5 6 7
|
3jca |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) ) |
9 |
1 2 3
|
rspc3v |
⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) → ( ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐸 ∀ 𝑧 ∈ 𝐹 𝜓 → 𝜒 ) ) |
10 |
8 4 9
|
sylc |
⊢ ( 𝜑 → 𝜒 ) |