Description: Obsolete version of rspn0 as of 28-Jun-2024. (Contributed by Alexander van der Vekens, 6-Sep-2018) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | rspn0OLD | ⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 | ⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 ) | |
| 2 | nfra1 | ⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝜑 | |
| 3 | nfv | ⊢ Ⅎ 𝑥 𝜑 | |
| 4 | 2 3 | nfim | ⊢ Ⅎ 𝑥 ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) |
| 5 | rsp | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) | |
| 6 | 5 | com12 | ⊢ ( 𝑥 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |
| 7 | 4 6 | exlimi | ⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |
| 8 | 1 7 | sylbi | ⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |