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Theorem nfeudw 2673
Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2676. Version of nfeud 2674 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfeudw.1 𝑦𝜑
nfeudw.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeudw (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfeudw
StepHypRef Expression
1 df-eu 2650 . 2 (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2 nfeudw.1 . . . 4 𝑦𝜑
3 nfeudw.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
42, 3nfexd 2344 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfmodv 2639 . . 3 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
64, 5nfand 1894 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
71, 6nfxfrd 1850 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wex 1776  wnf 1780  ∃*wmo 2616  ∃!weu 2649
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-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-mo 2618  df-eu 2650
This theorem is referenced by:  nfeuw  2675  nfreuw  3374
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