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Theorem nfmod2 2638
Description: Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2386. See nfmodv 2639 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2386. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2650. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfmod2.1 𝑦𝜑
nfmod2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfmod2 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Proof of Theorem nfmod2
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 nfmod2.1 . . . 4 𝑦𝜑
4 nfmod2.2 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5 nfeqf1 2393 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
65adantl 484 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
74, 6nfimd 1891 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
83, 7nfald2 2463 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
92, 8nfexd 2344 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
101, 9nfxfrd 1850 1 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1531  wex 1776  wnf 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  ax-13 2386
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-mo 2618
This theorem is referenced by:  nfmod  2641  nfeud2  2672  nfrmod  3373  nfrmo  3377  nfdisj  5043
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