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)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | reueq1 | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq2 | ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) ) | |
| 2 | 1 | anbi1d | ⊢ ( 𝐴 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ) |
| 3 | 2 | eubidv | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ) |
| 4 | df-reu | ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) | |
| 5 | df-reu | ⊢ ( ∃! 𝑥 ∈ 𝐵 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) | |
| 6 | 3 4 5 | 3bitr4g | ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) |