Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018) Avoid ax-10 , ax-12 . (Revised by GG, 28-Jun-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | rspn0 | ⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 | ⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 ) | |
| 2 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) | |
| 3 | exim | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 𝜑 ) ) | |
| 4 | ax5e | ⊢ ( ∃ 𝑥 𝜑 → 𝜑 ) | |
| 5 | 3 4 | syl6com | ⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → 𝜑 ) ) |
| 6 | 2 5 | biimtrid | ⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |
| 7 | 1 6 | sylbi | ⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) ) |