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Metamath Proof Explorer
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997) (Proof shortened by Wolf Lammen, 1-Dec-2019)
|
|
Ref |
Expression |
|
Assertion |
ralim |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
id |
⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) |
| 2 |
1
|
ral2imi |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |