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Theorem euabex 4157
 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 3601 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 vex 2693 . . . . 5 𝑦 ∈ V
32snex 4118 . . . 4 {𝑦} ∈ V
4 eleq1 2203 . . . 4 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} ∈ V ↔ {𝑦} ∈ V))
53, 4mpbiri 167 . . 3 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
65exlimiv 1578 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
71, 6sylbi 120 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃!weu 2000  {cab 2126  Vcvv 2690  {csn 3533 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539 This theorem is referenced by: (None)
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