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Mirrors > Home > MPE Home > Th. List > nfreuw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3432 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2372. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2117, ax-ext 2704. (Revised by Wolf Lammen, 21-Nov-2024.) |
Ref | Expression |
---|---|
nfrmow.1 | ⊢ Ⅎ𝑥𝐴 |
nfrmow.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfreuw | ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3378 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | nfrmow.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 3, 4 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
6 | 5 | nfeuw 2588 | . 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑) |
7 | 1, 6 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 Ⅎwnf 1786 ∈ wcel 2107 ∃!weu 2563 Ⅎwnfc 2884 ∃!wreu 3375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-mo 2535 df-eu 2564 df-clel 2811 df-nfc 2886 df-reu 3378 |
This theorem is referenced by: sbcreu 3871 reuccatpfxs1 14697 2reu7 45819 2reu8 45820 |
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