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Theorem nfreuw 3414
Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3435 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2118, ax-ext 2708. (Revised by Wolf Lammen, 21-Nov-2024.)
Hypotheses
Ref Expression
nfrmow.1 𝑥𝐴
nfrmow.2 𝑥𝜑
Assertion
Ref Expression
nfreuw 𝑥∃!𝑦𝐴 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfreuw
StepHypRef Expression
1 df-reu 3381 . 2 (∃!𝑦𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐴𝜑))
2 nfrmow.1 . . . . 5 𝑥𝐴
32nfcri 2897 . . . 4 𝑥 𝑦𝐴
4 nfrmow.2 . . . 4 𝑥𝜑
53, 4nfan 1899 . . 3 𝑥(𝑦𝐴𝜑)
65nfeuw 2593 . 2 𝑥∃!𝑦(𝑦𝐴𝜑)
71, 6nfxfr 1853 1 𝑥∃!𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  wa 395  wnf 1783  wcel 2108  ∃!weu 2568  wnfc 2890  ∃!wreu 3378
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-8 2110  ax-10 2141  ax-11 2157  ax-12 2177
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  df-clel 2816  df-nfc 2892  df-reu 3381
This theorem is referenced by:  sbcreu  3876  reuccatpfxs1  14785  2reu7  47123  2reu8  47124
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