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Theorem nfreuw 3412
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.)
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 3379 . 2 (∃!𝑦𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐴𝜑))
2 nfrmow.1 . . . . 5 𝑥𝐴
32nfcri 2895 . . . 4 𝑥 𝑦𝐴
4 nfrmow.2 . . . 4 𝑥𝜑
53, 4nfan 1897 . . 3 𝑥(𝑦𝐴𝜑)
65nfeuw 2591 . 2 𝑥∃!𝑦(𝑦𝐴𝜑)
71, 6nfxfr 1850 1 𝑥∃!𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  wa 395  wnf 1780  wcel 2106  ∃!weu 2566  wnfc 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
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