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Mirrors > Home > MPE Home > Th. List > ax9 | Structured version Visualization version GIF version |
Description: Proof of ax-9 1986 from ax9v1 1988 and ax9v2 1989, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 1987, which is itself a weakened version of ax-9 1986. (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 1946 | . 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦)) | |
2 | ax9v2 1989 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡)) | |
3 | 2 | equcoms 1934 | . . . 4 ⊢ (𝑡 = 𝑥 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡)) |
4 | ax9v1 1988 | . . . 4 ⊢ (𝑡 = 𝑦 → (𝑧 ∈ 𝑡 → 𝑧 ∈ 𝑦)) | |
5 | 3, 4 | sylan9 687 | . . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
6 | 5 | exlimiv 1845 | . 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
7 | 1, 6 | sylbi 206 | 1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: elequ2 1991 el 4773 dtru 4783 fv3 6116 elirrv 8387 bj-ax89 31854 bj-el 31984 bj-dtru 31985 axc11next 37629 |
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