Description: Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018) (Proof shortened by AV, 13-Oct-2018)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ralralimp | ⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → ( 𝜃 ∨ 𝜏 ) ) ∧ ¬ 𝜃 ) → 𝜏 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ornld | ⊢ ( 𝜑 → ( ( ( 𝜑 → ( 𝜃 ∨ 𝜏 ) ) ∧ ¬ 𝜃 ) → 𝜏 ) ) | |
| 2 | 1 | adantr | ⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( ( ( 𝜑 → ( 𝜃 ∨ 𝜏 ) ) ∧ ¬ 𝜃 ) → 𝜏 ) ) |
| 3 | 2 | ralimdv | ⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → ( 𝜃 ∨ 𝜏 ) ) ∧ ¬ 𝜃 ) → ∀ 𝑥 ∈ 𝐴 𝜏 ) ) |
| 4 | rspn0 | ⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝜏 → 𝜏 ) ) | |
| 5 | 4 | adantl | ⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( ∀ 𝑥 ∈ 𝐴 𝜏 → 𝜏 ) ) |
| 6 | 3 5 | syld | ⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → ( 𝜃 ∨ 𝜏 ) ) ∧ ¬ 𝜃 ) → 𝜏 ) ) |