Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Avoid ax-8 . (Revised by Wolf Lammen, 12-Mar-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | reueq1 | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexeq | ⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) ) | |
| 2 | rmoeq1 | ⊢ ( 𝐴 = 𝐵 → ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) | |
| 3 | 1 2 | anbi12d | ⊢ ( 𝐴 = 𝐵 → ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∃ 𝑥 ∈ 𝐵 𝜑 ∧ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) ) |
| 4 | reu5 | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) ) | |
| 5 | reu5 | ⊢ ( ∃! 𝑥 ∈ 𝐵 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐵 𝜑 ∧ ∃* 𝑥 ∈ 𝐵 𝜑 ) ) | |
| 6 | 3 4 5 | 3bitr4g | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) |