| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > r19.21bi | Structured version Visualization version GIF version | ||
| Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 11-Jun-2023.) |
| Ref | Expression |
|---|---|
| r19.21bi.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
| Ref | Expression |
|---|---|
| r19.21bi | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.21bi.1 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | |
| 2 | rspa 3248 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
| 3 | 1, 2 | sylan 580 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
| Copyright terms: Public domain | W3C validator |