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| Mirrors > Home > MPE Home > Th. List > nfmodv | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2560 for a version without disjoint variable conditions but requiring ax-13 2376. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.) | 
| Ref | Expression | 
|---|---|
| nfmodv.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfmodv.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfmodv | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-mo 2539 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
| 2 | nfv 1913 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
| 3 | nfmodv.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 4 | nfmodv.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 5 | nfvd 1914 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
| 6 | 4, 5 | nfimd 1893 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧)) | 
| 7 | 3, 6 | nfald 2327 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧)) | 
| 8 | 2, 7 | nfexd 2328 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | 
| 9 | 1, 8 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 ∃wex 1778 Ⅎwnf 1782 ∃*wmo 2537 | 
| 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-or 848 df-ex 1779 df-nf 1783 df-mo 2539 | 
| This theorem is referenced by: nfmov 2559 nfeudw 2590 nfrmowOLD 3426 nfdisjw 5121 wl-mo3t 37578 | 
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