Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995) (Revised by Gino Giotto, 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 | df-mo | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ) | |
3 | equequ1 | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 ↔ 𝑦 = 𝑧 ) ) | |
4 | 1 3 | imbi12d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝑥 = 𝑧 ) ↔ ( 𝜓 → 𝑦 = 𝑧 ) ) ) |
5 | 4 | cbvalvw | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ↔ ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ) |
6 | 5 | exbii | ⊢ ( ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ) |
7 | df-mo | ⊢ ( ∃* 𝑦 𝜓 ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ) | |
8 | 7 | bicomi | ⊢ ( ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) ↔ ∃* 𝑦 𝜓 ) |
9 | 2 6 8 | 3bitri | ⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 ) |