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Theorem euabex 3961
 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 3439 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 vex 2560 . . . . 5 𝑦 ∈ V
3 snexgOLD 3935 . . . . 5 (𝑦 ∈ V → {𝑦} ∈ V)
42, 3ax-mp 7 . . . 4 {𝑦} ∈ V
5 eleq1 2100 . . . 4 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} ∈ V ↔ {𝑦} ∈ V))
64, 5mpbiri 157 . . 3 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
76exlimiv 1489 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
81, 7sylbi 114 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900  {cab 2026  Vcvv 2557  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by: (None)
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