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Mirrors > Home > MPE Home > Th. List > equviniOLD | Structured version Visualization version GIF version |
Description: Obsolete version of equvini 2477 as of 16-Sep-2023. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
equviniOLD | ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtr 2028 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦)) | |
2 | equeuclr 2030 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) | |
3 | 2 | anc2ri 559 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
4 | 1, 3 | syli 39 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
5 | 19.8a 2180 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) | |
6 | 4, 5 | syl6 35 | . 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
7 | ax13 2393 | . . 3 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
8 | ax6e 2401 | . . . . 5 ⊢ ∃𝑧 𝑧 = 𝑦 | |
9 | 8, 3 | eximii 1837 | . . . 4 ⊢ ∃𝑧(𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
10 | 9 | 19.35i 1879 | . . 3 ⊢ (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
11 | 7, 10 | syl6 35 | . 2 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
12 | 6, 11 | pm2.61i 184 | 1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀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-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: (None) |
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