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Theorem nfeudv 1963
Description: Deduction version of nfeu 1967. Similar to nfeud 1964 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 1466 . . 3 𝑧𝜑
2 nfeudv.1 . . . 4 𝑦𝜑
3 nfeudv.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
4 nfv 1466 . . . . . 6 𝑥 𝑦 = 𝑧
54a1i 9 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
63, 5nfbid 1525 . . . 4 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
72, 6nfald 1690 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
81, 7nfexd 1691 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
9 df-eu 1951 . . 3 (∃!𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
109nfbii 1407 . 2 (Ⅎ𝑥∃!𝑦𝜓 ↔ Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
118, 10sylibr 132 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wnf 1394  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-eu 1951
This theorem is referenced by:  nfeud  1964
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