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Theorem hbeud 1938
 Description: Deduction version of hbeu 1937. (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 1367 . . 3 𝑦𝜑
3 hbeud.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
43nfi 1367 . . . 4 𝑥𝜑
5 hbeud.3 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
64, 5nfd 1432 . . 3 (𝜑 → Ⅎ𝑥𝜓)
72, 6nfeud 1932 . 2 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
87nfrd 1429 1 (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1257  ∃!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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919 This theorem is referenced by: (None)
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