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Mirrors > Home > MPE Home > Th. List > ax13lem1 | Structured version Visualization version GIF version |
Description: A version of ax13v 2373 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2375 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax13lem1 | ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equvinva 2033 | . 2 ⊢ (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤 ∧ 𝑦 = 𝑤)) | |
2 | ax13v 2373 | . . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤)) | |
3 | equeucl 2027 | . . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑦 = 𝑤 → 𝑧 = 𝑦)) | |
4 | 3 | alimdv 1919 | . . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)) |
5 | 2, 4 | syl9 77 | . . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))) |
6 | 5 | impd 411 | . . 3 ⊢ (¬ 𝑥 = 𝑦 → ((𝑧 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦)) |
7 | 6 | exlimdv 1936 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑤(𝑧 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦)) |
8 | 1, 7 | syl5 34 | 1 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀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-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 |
This theorem is referenced by: ax13 2375 ax13lem2 2376 nfeqf2 2377 ax6e 2383 wl-19.8eqv 35690 wl-19.2reqv 35691 |
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