Description: Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004) (Revised by Andrew Salmon, 11-Jul-2011)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rmoeq1f.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| rmoeq1f.2 | ⊢ Ⅎ 𝑥 𝐵 | ||
| Assertion | reueq1f | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rmoeq1f.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| 2 | rmoeq1f.2 | ⊢ Ⅎ 𝑥 𝐵 | |
| 3 | 1 2 | rexeqf | ⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) ) |
| 4 | 1 2 | rmoeq1f | ⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) |
| 5 | 3 4 | anbi12d | ⊢ ( 𝐴 = 𝐵 → ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∃ 𝑥 ∈ 𝐵 𝜑 ∧ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) ) |
| 6 | reu5 | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) ) | |
| 7 | reu5 | ⊢ ( ∃! 𝑥 ∈ 𝐵 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐵 𝜑 ∧ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) | |
| 8 | 5 6 7 | 3bitr4g | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) |