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Theorem nfmod 2023
 Description: Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
nfeud.1 𝑦𝜑
nfeud.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfmod (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Proof of Theorem nfmod
StepHypRef Expression
1 df-mo 2010 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2 nfeud.1 . . . 4 𝑦𝜑
3 nfeud.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
42, 3nfexd 1741 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfeud 2022 . . 3 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
64, 5nfimd 1565 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
71, 6nfxfrd 1455 1 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1440  ∃wex 1472  ∃!weu 2006  ∃*wmo 2007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010 This theorem is referenced by:  nfmo  2026
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