Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 9-Mar-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvraldva.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| Assertion | cbvraldva | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvraldva.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | 1 | ancoms | ⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝜓 ↔ 𝜒 ) ) |
| 3 | 2 | pm5.74da | ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) ) |
| 4 | 3 | cbvralvw | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) ) |
| 5 | r19.21v | ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) | |
| 6 | r19.21v | ⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 𝜒 ) ) | |
| 7 | 4 5 6 | 3bitr3i | ⊢ ( ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 𝜒 ) ) |
| 8 | 7 | pm5.74ri | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 𝜒 ) ) |