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Mirrors > Home > MPE Home > Th. List > ax9 | Structured version Visualization version GIF version |
Description: Proof of ax-9 2091 from ax9v1 2093 and ax9v2 2094, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2092, which is itself a weakened version of ax-9 2091. (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 2013 | . 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦)) | |
2 | ax9v2 2094 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡)) | |
3 | 2 | equcoms 2004 | . . . 4 ⊢ (𝑡 = 𝑥 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡)) |
4 | ax9v1 2093 | . . . 4 ⊢ (𝑡 = 𝑦 → (𝑧 ∈ 𝑡 → 𝑧 ∈ 𝑦)) | |
5 | 3, 4 | sylan9 508 | . . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
6 | 5 | exlimiv 1908 | . 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
7 | 1, 6 | sylbi 218 | 1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-9 2091 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 |
This theorem is referenced by: elequ2 2096 el 5161 fv3 6556 elirrv 8906 bj-ax89 33610 bj-el 33703 bj-dtru 33704 axc11next 40276 |
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