This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Inference adding restricted universal quantifier to both sides of an
equivalence. (Contributed by NM, 26-Nov-2000)
|
|
Ref |
Expression |
|
Hypothesis |
ralbiia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
ralbiia |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ralbiia.1 |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
| 2 |
1
|
pm5.74i |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → 𝜓 ) ) |
| 3 |
2
|
ralbii2 |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 ) |