Description: Obsolete version of euim as of 1-Oct-2023. (Contributed by NM, 19-Oct-2005) (Proof shortened by Andrew Salmon, 14-Jun-2011) (New usage is discouraged.) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | euimOLD | ⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → ( ∃! 𝑥 𝜓 → ∃! 𝑥 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1 | ⊢ ( ∃ 𝑥 𝜑 → ( ∃! 𝑥 𝜓 → ∃ 𝑥 𝜑 ) ) | |
| 2 | euimmo | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜑 ) ) | |
| 3 | 1 2 | anim12ii | ⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → ( ∃! 𝑥 𝜓 → ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) ) ) |
| 4 | df-eu | ⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) ) | |
| 5 | 3 4 | syl6ibr | ⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → ( ∃! 𝑥 𝜓 → ∃! 𝑥 𝜑 ) ) |