Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003) (Proof shortened by Wolf Lammen, 31-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ralbida.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| ralbida.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) | ||
| Assertion | ralbida | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralbida.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| 2 | ralbida.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 3 | 2 | biimpd | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) ) |
| 4 | 1 3 | ralimdaa | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |
| 5 | 2 | biimprd | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜒 → 𝜓 ) ) |
| 6 | 1 5 | ralimdaa | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜒 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) |
| 7 | 4 6 | impbid | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |