Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 8-Jan-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbvraldva2.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| cbvraldva2.2 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) | ||
| Assertion | cbvrexdva2 | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvraldva2.1 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | cbvraldva2.2 | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) | |
| 3 | 1 | notbid | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
| 4 | 3 2 | cbvraldva2 | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝜒 ) ) |
| 5 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ) | |
| 6 | ralnex | ⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝜒 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝜒 ) | |
| 7 | 4 5 6 | 3bitr3g | ⊢ ( 𝜑 → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝜒 ) ) |
| 8 | 7 | con4bid | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) ) |