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| Description: Proof of ax-9 2118 from ax9v1 2120 and ax9v2 2121, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2119, which is itself a weakened version of ax-9 2118. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| ax9 | ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equvinv 2028 | . 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦)) | |
| 2 | ax9v2 2121 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡)) | |
| 3 | 2 | equcoms 2019 | . . . 4 ⊢ (𝑡 = 𝑥 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡)) | 
| 4 | ax9v1 2120 | . . . 4 ⊢ (𝑡 = 𝑦 → (𝑧 ∈ 𝑡 → 𝑧 ∈ 𝑦)) | |
| 5 | 3, 4 | sylan9 507 | . . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | 
| 6 | 5 | exlimiv 1930 | . 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | 
| 7 | 1, 6 | sylbi 217 | 1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | 
| 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-9 2118 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 | 
| This theorem is referenced by: elequ2 2123 elALT2 5369 fv3 6924 elirrv 9636 in-ax8 36225 ss-ax8 36226 bj-ax89 36679 axc11next 44425 | 
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