Metamath Proof Explorer
Description: Formula-building rule for restricted universal quantifiers (deduction
form.) (Contributed by Scott Fenton, 20-Feb-2025)
|
|
Ref |
Expression |
|
Hypothesis |
3ralbidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
3ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜒 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
3ralbidv.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
2 |
1
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝐶 𝜓 ↔ ∀ 𝑧 ∈ 𝐶 𝜒 ) ) |
3 |
2
|
2ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜒 ) ) |