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Mirrors > Home > MPE Home > Th. List > ax8 | Structured version Visualization version GIF version |
Description: Proof of ax-8 2109 from ax8v1 2111 and ax8v2 2112, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2110, which is itself a weakened version of ax-8 2109. (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 2033 | . 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦)) | |
2 | ax8v2 2112 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧)) | |
3 | 2 | equcoms 2024 | . . . 4 ⊢ (𝑡 = 𝑥 → (𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧)) |
4 | ax8v1 2111 | . . . 4 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
5 | 3, 4 | sylan9 509 | . . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
6 | 5 | exlimiv 1934 | . 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: elequ1 2114 ax9ALT 2728 elALT2 5368 axextdfeq 34769 ax8dfeq 34770 exnel 34774 bj-ax89 35555 |
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