Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017) (Revised by Gino Giotto, 10-Jan-2024) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvaldvaw.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| Assertion | cbvaldvaw | ⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvaldvaw.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 | cbvalvw | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ( 𝜑 → 𝜒 ) ) |
| 5 | 19.21v | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) | |
| 6 | 19.21v | ⊢ ( ∀ 𝑦 ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → ∀ 𝑦 𝜒 ) ) | |
| 7 | 4 5 6 | 3bitr3i | ⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜒 ) ) |
| 8 | 7 | pm5.74ri | ⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) ) |