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Theorem euabex 4117
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 3562 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 vex 2663 . . . . 5 𝑦 ∈ V
32snex 4079 . . . 4 {𝑦} ∈ V
4 eleq1 2180 . . . 4 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} ∈ V ↔ {𝑦} ∈ V))
53, 4mpbiri 167 . . 3 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
65exlimiv 1562 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
71, 6sylbi 120 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wex 1453  wcel 1465  ∃!weu 1977  {cab 2103  Vcvv 2660  {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503
This theorem is referenced by: (None)
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