Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo and cbvmow for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995) (Revised by GG, 30-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvmovw.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | cbvmovw | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvmovw.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | equequ1 | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 ↔ 𝑦 = 𝑧 ) ) | |
| 3 | 1 2 | imbi12d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝑥 = 𝑧 ) ↔ ( 𝜓 → 𝑦 = 𝑧 ) ) ) |
| 4 | 3 | cbvalvw | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ↔ ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ) |
| 5 | 4 | exbii | ⊢ ( ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ) |
| 6 | df-mo | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ) | |
| 7 | df-mo | ⊢ ( ∃* 𝑦 𝜓 ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ) | |
| 8 | 5 6 7 | 3bitr4i | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 ) |