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Theorem nfmodv 2557
Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2559 for a version without disjoint variable conditions but requiring ax-13 2375. (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 2538 . 2 (∃*𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
2 nfv 1912 . . 3 𝑧𝜑
3 nfmodv.1 . . . 4 𝑦𝜑
4 nfmodv.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
5 nfvd 1913 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
64, 5nfimd 1892 . . . 4 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
73, 6nfald 2327 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
82, 7nfexd 2328 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
91, 8nfxfrd 1851 1 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wex 1776  wnf 1780  ∃*wmo 2536
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-or 848  df-ex 1777  df-nf 1781  df-mo 2538
This theorem is referenced by:  nfmov  2558  nfeudw  2589  nfrmowOLD  3424  nfdisjw  5127  wl-mo3t  37557
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