Metamath Proof Explorer
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007)
|
|
Ref |
Expression |
|
Hypothesis |
ralimiaa.1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) |
|
Assertion |
ralimiaa |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ralimiaa.1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) |
2 |
1
|
ex |
⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) |
3 |
2
|
ralimia |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) |