Description: Rule used to change the bound variable in a restricted existential 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 | cbvrexdva | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐴 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvraldva.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | 1 | notbid | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
| 3 | 2 | cbvraldva | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 ¬ 𝜒 ) ) |
| 4 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ) | |
| 5 | ralnex | ⊢ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝜒 ↔ ¬ ∃ 𝑦 ∈ 𝐴 𝜒 ) | |
| 6 | 3 4 5 | 3bitr3g | ⊢ ( 𝜑 → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃ 𝑦 ∈ 𝐴 𝜒 ) ) |
| 7 | 6 | con4bid | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐴 𝜒 ) ) |