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Metamath Proof Explorer
Description: Inference adding three restricted universal quantifiers to both sides of
an equivalence. (Contributed by Peter Mazsa, 25-Jul-2019)
|
|
Ref |
Expression |
|
Hypothesis |
3ralbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
|
Assertion |
3ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
3ralbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
| 2 |
1
|
2ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ) |
| 3 |
2
|
ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 ) |