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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 2593. Version of nfeud 2591 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 15-Feb-2013.) Avoid ax-13 2376. (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 2568 | . 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | |
| 2 | nfeudw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfeudw.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 4 | 2, 3 | nfexd 2328 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | 
| 5 | 2, 3 | nfmodv 2558 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | 
| 6 | 4, 5 | nfand 1896 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | 
| 7 | 1, 6 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1778 Ⅎwnf 1782 ∃*wmo 2537 ∃!weu 2567 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: nfeuw 2592 nfreuwOLD 3425 | 
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