Description: Obsolete version of ralbida as of 31-Oct-2024. (Contributed by NM, 6-Oct-2003) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ralbida.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| ralbida.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) | ||
| Assertion | ralbidaOLD | ⊢ ( 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 | pm5.74da | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 → 𝜒 ) ) ) |
| 4 | 1 3 | albid | ⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) ) ) |
| 5 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) | |
| 6 | df-ral | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) ) | |
| 7 | 4 5 6 | 3bitr4g | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) ) |