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Theorem hbeud 1970
Description: Deduction version of hbeu 1969. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
Hypotheses
Ref Expression
hbeud.1 (𝜑 → ∀𝑥𝜑)
hbeud.2 (𝜑 → ∀𝑦𝜑)
hbeud.3 (𝜑 → (𝜓 → ∀𝑥𝜓))
Assertion
Ref Expression
hbeud (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))

Proof of Theorem hbeud
StepHypRef Expression
1 hbeud.2 . . . 4 (𝜑 → ∀𝑦𝜑)
21nfi 1396 . . 3 𝑦𝜑
3 hbeud.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
43nfi 1396 . . . 4 𝑥𝜑
5 hbeud.3 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
64, 5nfd 1461 . . 3 (𝜑 → Ⅎ𝑥𝜓)
72, 6nfeud 1964 . 2 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
87nfrd 1458 1 (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  ∃!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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by: (None)
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