Description: Proof of ax-8 from ax8v1 and ax8v2 , proving sufficiency of the conjunction of the latter two weakened versions of ax8v , which is itself a weakened version of ax-8 . (Contributed by BJ, 7-Dec-2020) (Proof shortened by Wolf Lammen, 11-Apr-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax8 | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equvinv | ⊢ ( 𝑥 = 𝑦 ↔ ∃ 𝑡 ( 𝑡 = 𝑥 ∧ 𝑡 = 𝑦 ) ) | |
| 2 | ax8v2 | ⊢ ( 𝑥 = 𝑡 → ( 𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧 ) ) | |
| 3 | 2 | equcoms | ⊢ ( 𝑡 = 𝑥 → ( 𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧 ) ) |
| 4 | ax8v1 | ⊢ ( 𝑡 = 𝑦 → ( 𝑡 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) | |
| 5 | 3 4 | sylan9 | ⊢ ( ( 𝑡 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) |
| 6 | 5 | exlimiv | ⊢ ( ∃ 𝑡 ( 𝑡 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) |
| 7 | 1 6 | sylbi | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) |