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Theorem nfmod2 2471
 Description: Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
nfmod2.1 𝑦𝜑
nfmod2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfmod2 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Proof of Theorem nfmod2
StepHypRef Expression
1 df-mo 2463 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2 nfmod2.1 . . . 4 𝑦𝜑
3 nfmod2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
42, 3nfexd2 2320 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfeud2 2470 . . 3 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
64, 5nfimd 1812 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
71, 6nfxfrd 1772 1 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  ∃!weu 2458  ∃*wmo 2459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463 This theorem is referenced by:  nfmod  2473  nfrmod  3092  nfrmo  3094  nfdisj  4565
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