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| Mirrors > Home > MPE Home > Th. List > ax8 | Structured version Visualization version GIF version | ||
| Description: Proof of ax-8 2110 from ax8v1 2112 and ax8v2 2113, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2111, which is itself a weakened version of ax-8 2110. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.) |
| Ref | Expression |
|---|---|
| ax8 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equvinv 2028 | . 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦)) | |
| 2 | ax8v2 2113 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧)) | |
| 3 | 2 | equcoms 2019 | . . . 4 ⊢ (𝑡 = 𝑥 → (𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧)) |
| 4 | ax8v1 2112 | . . . 4 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
| 5 | 3, 4 | sylan9 507 | . . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
| 6 | 5 | exlimiv 1930 | . 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
| 7 | 1, 6 | sylbi 217 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: elequ1 2115 ax9ALT 2732 elALT2 5369 axextdfeq 35798 ax8dfeq 35799 exnel 35803 bj-ax89 36679 |
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