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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax6e | Structured version Visualization version GIF version |
Description: Proof of ax6e 2383 (hence ax6 2384) from Tarski's system, ax-c9 36891, ax-c16 36893. Remark: ax-6 1971 is used only via its principal (unbundled) instance ax6v 1972. (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 1980 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦) | |
2 | 1 | a1d 25 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)) |
3 | bj-ax6elem1 34834 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | |
4 | bj-ax6elem2 34835 | . . . 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 182 | . 2 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) |
7 | ax6evr 2018 | . 2 ⊢ ∃𝑧 𝑦 = 𝑧 | |
8 | 6, 7 | exlimiiv 1934 | 1 ⊢ ∃𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 ∃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 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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