Metamath Proof Explorer
Description: Triple application of rspcedvdw . (Contributed by SN, 20-Aug-2024)
|
|
Ref |
Expression |
|
Hypotheses |
3rspcedvdw.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) |
|
|
3rspcedvdw.2 |
⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) ) |
|
|
3rspcedvdw.3 |
⊢ ( 𝑧 = 𝐶 → ( 𝜃 ↔ 𝜏 ) ) |
|
|
3rspcedvdw.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
|
|
3rspcedvdw.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑌 ) |
|
|
3rspcedvdw.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑍 ) |
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|
3rspcedvdw.4 |
⊢ ( 𝜑 → 𝜏 ) |
|
Assertion |
3rspcedvdw |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑌 ∃ 𝑧 ∈ 𝑍 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
3rspcedvdw.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) |
2 |
|
3rspcedvdw.2 |
⊢ ( 𝑦 = 𝐵 → ( 𝜒 ↔ 𝜃 ) ) |
3 |
|
3rspcedvdw.3 |
⊢ ( 𝑧 = 𝐶 → ( 𝜃 ↔ 𝜏 ) ) |
4 |
|
3rspcedvdw.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
5 |
|
3rspcedvdw.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑌 ) |
6 |
|
3rspcedvdw.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑍 ) |
7 |
|
3rspcedvdw.4 |
⊢ ( 𝜑 → 𝜏 ) |
8 |
1 2 3
|
rspc3ev |
⊢ ( ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ) ∧ 𝜏 ) → ∃ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑌 ∃ 𝑧 ∈ 𝑍 𝜓 ) |
9 |
4 5 6 7 8
|
syl31anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝑌 ∃ 𝑧 ∈ 𝑍 𝜓 ) |