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

Theorem nfreuw 3374
 Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3376 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfreuw.1 𝑥𝐴
nfreuw.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 3145 . . 3 (∃!𝑦𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐴𝜑))
2 nftru 1801 . . . 4 𝑦
3 nfcvd 2978 . . . . . 6 (⊤ → 𝑥𝑦)
4 nfreuw.1 . . . . . . 7 𝑥𝐴
54a1i 11 . . . . . 6 (⊤ → 𝑥𝐴)
63, 5nfeld 2989 . . . . 5 (⊤ → Ⅎ𝑥 𝑦𝐴)
7 nfreuw.2 . . . . . 6 𝑥𝜑
87a1i 11 . . . . 5 (⊤ → Ⅎ𝑥𝜑)
96, 8nfand 1894 . . . 4 (⊤ → Ⅎ𝑥(𝑦𝐴𝜑))
102, 9nfeudw 2673 . . 3 (⊤ → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜑))
111, 10nfxfrd 1850 . 2 (⊤ → Ⅎ𝑥∃!𝑦𝐴 𝜑)
1211mptru 1540 1 𝑥∃!𝑦𝐴 𝜑
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 398  ⊤wtru 1534  Ⅎwnf 1780   ∈ wcel 2110  ∃!weu 2649  Ⅎwnfc 2961  ∃!wreu 3140 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-mo 2618  df-eu 2650  df-cleq 2814  df-clel 2893  df-nfc 2963  df-reu 3145 This theorem is referenced by:  sbcreu  3858  reuccatpfxs1  14108  2reu7  43309  2reu8  43310
 Copyright terms: Public domain W3C validator