Description: Proof of ax-9 from ax9v1 and ax9v2 , proving sufficiency of the conjunction of the latter two weakened versions of ax9v , which is itself a weakened version of ax-9 . (Contributed by BJ, 7-Dec-2020) (Proof shortened by Wolf Lammen, 11-Apr-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax9 | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equvinv | ⊢ ( 𝑥 = 𝑦 ↔ ∃ 𝑡 ( 𝑡 = 𝑥 ∧ 𝑡 = 𝑦 ) ) | |
| 2 | ax9v2 | ⊢ ( 𝑥 = 𝑡 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡 ) ) | |
| 3 | 2 | equcoms | ⊢ ( 𝑡 = 𝑥 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡 ) ) |
| 4 | ax9v1 | ⊢ ( 𝑡 = 𝑦 → ( 𝑧 ∈ 𝑡 → 𝑧 ∈ 𝑦 ) ) | |
| 5 | 3 4 | sylan9 | ⊢ ( ( 𝑡 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) ) |
| 6 | 5 | exlimiv | ⊢ ( ∃ 𝑡 ( 𝑡 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) ) |
| 7 | 1 6 | sylbi | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦 ) ) |