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Theorem euabex 4015
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
euabex (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)

Proof of Theorem euabex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3485 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 vex 2615 . . . . 5 𝑦 ∈ V
32snex 3983 . . . 4 {𝑦} ∈ V
4 eleq1 2145 . . . 4 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} ∈ V ↔ {𝑦} ∈ V))
53, 4mpbiri 166 . . 3 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
65exlimiv 1530 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
71, 6sylbi 119 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wex 1422  wcel 1434  ∃!weu 1943  {cab 2069  Vcvv 2612  {csn 3422
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428
This theorem is referenced by: (None)
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