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Theorem nfmodv 2639
 Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2641 for a version without disjoint variable conditions but requiring ax-13 2386. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.)
Hypotheses
Ref Expression
nfmodv.1 𝑦𝜑
nfmodv.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfmodv (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfmodv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mo 2618 . 2 (∃*𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
2 nfv 1911 . . 3 𝑧𝜑
3 nfmodv.1 . . . 4 𝑦𝜑
4 nfmodv.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
5 nfvd 1912 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
64, 5nfimd 1891 . . . 4 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
73, 6nfald 2343 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
82, 7nfexd 2344 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
91, 8nfxfrd 1850 1 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776  Ⅎwnf 1780  ∃*wmo 2616 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-or 844  df-ex 1777  df-nf 1781  df-mo 2618 This theorem is referenced by:  nfmov  2640  nfeudw  2673  nfrmow  3375  nfdisjw  5042
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