Description: Theorem 19.2 of Margaris p. 89 with restricted quantifiers (compare 19.2 ). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r19.2z | ⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝜑 ) → ∃ 𝑥 ∈ 𝐴 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) | |
| 2 | exintr | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) | |
| 3 | 1 2 | sylbi | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ) |
| 4 | n0 | ⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 ) | |
| 5 | df-rex | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) | |
| 6 | 3 4 5 | 3imtr4g | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 𝜑 ) ) |
| 7 | 6 | impcom | ⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝜑 ) → ∃ 𝑥 ∈ 𝐴 𝜑 ) |