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| Mirrors > Home > MPE Home > Th. List > nfeudw | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2594. Version of nfeud 2592 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 15-Feb-2013.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| nfeudw.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfeudw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfeudw | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-eu 2569 | . 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | |
| 2 | nfeudw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfeudw.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 4 | 2, 3 | nfexd 2329 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | 
| 5 | 2, 3 | nfmodv 2559 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | 
| 6 | 4, 5 | nfand 1897 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | 
| 7 | 1, 6 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 Ⅎwnf 1783 ∃*wmo 2538 ∃!weu 2568 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: nfeuw 2593 nfreuwOLD 3426 | 
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