![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2375. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2706. (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 3379 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | nfrmow.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2895 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 3, 4 | nfan 1897 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
6 | 5 | nfeuw 2591 | . 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑) |
7 | 1, 6 | nfxfr 1850 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 Ⅎwnf 1780 ∈ wcel 2106 ∃!weu 2566 Ⅎwnfc 2888 ∃!wreu 3376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-mo 2538 df-eu 2567 df-clel 2814 df-nfc 2890 df-reu 3379 |
This theorem is referenced by: sbcreu 3885 reuccatpfxs1 14782 2reu7 47061 2reu8 47062 |
Copyright terms: Public domain | W3C validator |