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Theorem nfeudw 2621
Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2624. Version of nfeud 2622 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 15-Feb-2013.) Avoid ax-13 2406. (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 2599 . 2 (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2 nfeudw.1 . . . 4 𝑦𝜑
3 nfeudw.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
42, 3nfexd 2364 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfmodv 2589 . . 3 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
64, 5nfand 1930 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
71, 6nfxfrd 1887 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wa 401  wex 1812  wnf 1816  ∃*wmo 2567  ∃!weu 2598
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-mo 2569  df-eu 2599
This theorem is used by:  nfeuw  2623
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