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Theorem hbeu 1937
Description: Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)
Hypothesis
Ref Expression
hbeu.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbeu (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Proof of Theorem hbeu
StepHypRef Expression
1 hbeu.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nfi 1367 . . 3 𝑥𝜑
32nfeu 1935 . 2 𝑥∃!𝑦𝜑
43nfri 1428 1 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
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:  hbmo  1955  2eu7  2010
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