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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 ) |