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Theorem nfmod2 2481
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 2473 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2 nfmod2.1 . . . 4 𝑦𝜑
3 nfmod2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
42, 3nfexd2 2330 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfeud2 2480 . . 3 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
64, 5nfimd 1821 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
71, 6nfxfrd 1778 1 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1479  wex 1702  wnf 1706  ∃!weu 2468  ∃*wmo 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-eu 2472  df-mo 2473
This theorem is referenced by:  nfmod  2483  nfrmod  3108  nfrmo  3110  nfdisj  4623
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