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Metamath Proof Explorer
Description: Inference adding four restricted universal quantifiers to both sides of
an equivalence. (Contributed by Scott Fenton, 28-Feb-2025)
|
|
Ref |
Expression |
|
Hypothesis |
4ralbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
|
Assertion |
4ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
4ralbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
| 2 |
1
|
ralbii |
⊢ ( ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑤 ∈ 𝐷 𝜓 ) |
| 3 |
2
|
3ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 ) |