This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Inference adding two restricted universal quantifiers to both sides of
an equivalence. (Contributed by NM, 1-Aug-2004)
|
|
Ref |
Expression |
|
Hypothesis |
2ralbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
|
Assertion |
2ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
2ralbii.1 |
⊢ ( 𝜑 ↔ 𝜓 ) |
| 2 |
1
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) |
| 3 |
2
|
ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ) |