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Metamath Proof Explorer
Description: Formula-building rule for restricted universal quantifiers (deduction
form.) (Contributed by Scott Fenton, 5-Mar-2025)
|
|
Ref |
Expression |
|
Hypothesis |
6ralbidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
6ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
6ralbidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
| 2 |
1
|
2ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜓 ↔ ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜒 ) ) |
| 3 |
2
|
4ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 ∀ 𝑡 ∈ 𝐸 ∀ 𝑢 ∈ 𝐹 𝜒 ) ) |