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Theorem hbeud 2060
Description: Deduction version of hbeu 2059. (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 1473 . . 3 𝑦𝜑
3 hbeud.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
43nfi 1473 . . . 4 𝑥𝜑
5 hbeud.3 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
64, 5nfd 1534 . . 3 (𝜑 → Ⅎ𝑥𝜓)
72, 6nfeud 2054 . 2 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
87nfrd 1531 1 (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  ∃!weu 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041
This theorem is referenced by: (None)
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