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Theorem nfeud2 2615
Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)
Hypotheses
Ref Expression
nfeud2.1 𝑦𝜑
nfeud2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeud2 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 2607 . 2 (∃!𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
2 nfv 1988 . . 3 𝑧𝜑
3 nfeud2.1 . . . 4 𝑦𝜑
4 nfeud2.2 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5 nfeqf1 2440 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
65adantl 473 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
74, 6nfbid 1977 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
83, 7nfald2 2467 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
92, 8nfexd 2308 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
101, 9nfxfrd 1925 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1626  wex 1849  wnf 1853  ∃!weu 2603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-eu 2607
This theorem is referenced by:  nfmod2  2616  nfeud  2617  nfreud  3246
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