Description: Under a non-freeness hypothesis, the implication 19.38 can be strengthened to an equivalence. See also 19.38b . (Contributed by BJ, 3-Nov-2021) (Proof shortened by Wolf Lammen, 9-Jul-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 19.38a | ⊢ ( Ⅎ 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.38 | ⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) | |
| 2 | id | ⊢ ( Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 𝜑 ) | |
| 3 | 2 | nfrd | ⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ) |
| 4 | alim | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) | |
| 5 | 3 4 | syl9 | ⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) |
| 6 | 1 5 | impbid2 | ⊢ ( Ⅎ 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) |