Description: Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)
Ref | Expression | ||
---|---|---|---|
Hypothesis | expandral.1 | ⊢ ( 𝜑 ↔ 𝜓 ) | |
Assertion | expandral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expandral.1 | ⊢ ( 𝜑 ↔ 𝜓 ) | |
2 | 1 | ralbii | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 ) |
3 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) | |
4 | 2 3 | bitri | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) |