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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax6e | Structured version Visualization version GIF version |
Description: Proof of ax6e 2401 (hence ax6 2402) from Tarski's system, ax-c9 36028, ax-c16 36030. Remark: ax-6 1970 is used only via its principal (unbundled) instance ax6v 1971. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-ax6e | ⊢ ∃𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.2 1981 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦) | |
2 | 1 | a1d 25 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)) |
3 | bj-ax6elem1 34001 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | |
4 | bj-ax6elem2 34002 | . . . 4 ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) | |
5 | 3, 4 | syl6 35 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)) |
6 | 2, 5 | pm2.61i 184 | . 2 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) |
7 | ax6evr 2022 | . 2 ⊢ ∃𝑧 𝑦 = 𝑧 | |
8 | 6, 7 | exlimiiv 1932 | 1 ⊢ ∃𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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