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Mirrors > Home > MPE Home > Th. List > ax13lem2 | Structured version Visualization version GIF version |
Description: Lemma for nfeqf2 2349. This lemma is equivalent to ax13v 2345 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax13lem2 | ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax13lem1 2346 | . . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦)) | |
2 | equeucl 2008 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑧 = 𝑤)) | |
3 | 2 | eximi 1816 | . . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∃𝑥(𝑤 = 𝑦 → 𝑧 = 𝑤)) |
4 | 19.36v 1971 | . . . . 5 ⊢ (∃𝑥(𝑤 = 𝑦 → 𝑧 = 𝑤) ↔ (∀𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤)) | |
5 | 3, 4 | sylib 219 | . . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∀𝑥 𝑤 = 𝑦 → 𝑧 = 𝑤)) |
6 | 1, 5 | syl9 77 | . . 3 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑧 = 𝑤))) |
7 | 6 | alrimdv 1907 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑤(𝑤 = 𝑦 → 𝑧 = 𝑤))) |
8 | equequ2 2010 | . . 3 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
9 | 8 | equsalvw 1987 | . 2 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑧 = 𝑤) ↔ 𝑧 = 𝑦) |
10 | 7, 9 | syl6ib 252 | 1 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1520 ∃wex 1761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-13 2344 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 |
This theorem is referenced by: nfeqf2 2349 nfeqf2OLD 2350 wl-speqv 34314 wl-19.2reqv 34316 |
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