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| Mirrors > Home > MPE Home > Th. List > nfeud2 | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use nfeudw 2591 instead. (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfeud2.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfeud2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfeud2 | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-eu 2569 | . 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | |
| 2 | nfeud2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfeud2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
| 4 | 2, 3 | nfexd2 2451 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | 
| 5 | 2, 3 | nfmod2 2558 | . . 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: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 ∃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 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: nfeud 2592 nfreud 3433 | 
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