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Theorem nfeudw 2589
Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2592. Version of nfeud 2590 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 15-Feb-2013.) Avoid ax-13 2375. (Revised by GG, 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 2567 . 2 (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2 nfeudw.1 . . . 4 𝑦𝜑
3 nfeudw.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
42, 3nfexd 2328 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfmodv 2557 . . 3 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
64, 5nfand 1895 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
71, 6nfxfrd 1851 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wnf 1780  ∃*wmo 2536  ∃!weu 2566
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-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567
This theorem is referenced by:  nfeuw  2591  nfreuwOLD  3423
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