MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfreuw Structured version   Visualization version   GIF version

Theorem nfreuw 3411
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.)
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 3378 . 2 (∃!𝑦𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐴𝜑))
2 nfrmow.1 . . . . 5 𝑥𝐴
32nfcri 2891 . . . 4 𝑥 𝑦𝐴
4 nfrmow.2 . . . 4 𝑥𝜑
53, 4nfan 1903 . . 3 𝑥(𝑦𝐴𝜑)
65nfeuw 2588 . 2 𝑥∃!𝑦(𝑦𝐴𝜑)
71, 6nfxfr 1856 1 𝑥∃!𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  wa 397  wnf 1786  wcel 2107  ∃!weu 2563  wnfc 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
  Copyright terms: Public domain W3C validator