Description: Double existential uniqueness implies double unique existential quantification. Version of 2exeu with x and y distinct, but not requiring ax-13 . (Contributed by NM, 3-Dec-2001) (Revised by Wolf Lammen, 2-Oct-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 2exeuv | ⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) → ∃! 𝑥 ∃! 𝑦 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eumo | ⊢ ( ∃! 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑥 ∃ 𝑦 𝜑 ) | |
| 2 | euex | ⊢ ( ∃! 𝑦 𝜑 → ∃ 𝑦 𝜑 ) | |
| 3 | 2 | moimi | ⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑥 ∃! 𝑦 𝜑 ) |
| 4 | 1 3 | syl | ⊢ ( ∃! 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑥 ∃! 𝑦 𝜑 ) |
| 5 | 2euexv | ⊢ ( ∃! 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑥 ∃! 𝑦 𝜑 ) | |
| 6 | 4 5 | anim12ci | ⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) → ( ∃ 𝑥 ∃! 𝑦 𝜑 ∧ ∃* 𝑥 ∃! 𝑦 𝜑 ) ) |
| 7 | df-eu | ⊢ ( ∃! 𝑥 ∃! 𝑦 𝜑 ↔ ( ∃ 𝑥 ∃! 𝑦 𝜑 ∧ ∃* 𝑥 ∃! 𝑦 𝜑 ) ) | |
| 8 | 6 7 | sylibr | ⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) → ∃! 𝑥 ∃! 𝑦 𝜑 ) |