Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020) (Proof shortened by Wolf Lammen, 11-Dec-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r19.21v | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.27 | ⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) | |
| 2 | 1 | ralimdv | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |
| 3 | 2 | com12 | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |
| 4 | pm2.21 | ⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) ) | |
| 5 | 4 | ralrimivw | ⊢ ( ¬ 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ) |
| 6 | ax-1 | ⊢ ( 𝜓 → ( 𝜑 → 𝜓 ) ) | |
| 7 | 6 | ralimi | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ) |
| 8 | 5 7 | ja | ⊢ ( ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ) |
| 9 | 3 8 | impbii | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |