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Theorem nfeudv 1931
Description: Deduction version of nfeu 1935. Similar to nfeud 1932 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
Hypotheses
Ref Expression
nfeudv.1 𝑦𝜑
nfeudv.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeudv (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfeudv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . 3 𝑧𝜑
2 nfeudv.1 . . . 4 𝑦𝜑
3 nfeudv.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
4 nfv 1437 . . . . . 6 𝑥 𝑦 = 𝑧
54a1i 9 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
63, 5nfbid 1496 . . . 4 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
72, 6nfald 1659 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
81, 7nfexd 1660 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
9 df-eu 1919 . . 3 (∃!𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
109nfbii 1378 . 2 (Ⅎ𝑥∃!𝑦𝜓 ↔ Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
118, 10sylibr 141 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wnf 1365  wex 1397  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-eu 1919
This theorem is referenced by:  nfeud  1932
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