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Theorem moabex 4888
Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
Assertion
Ref Expression
moabex (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)

Proof of Theorem moabex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mo2v 2476 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abss 3650 . . . . 5 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4164 . . . . . . 7 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43imbi2i 326 . . . . . 6 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1744 . . . . 5 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 264 . . . 4 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
7 snex 4869 . . . . 5 {𝑦} ∈ V
87ssex 4762 . . . 4 ({𝑥𝜑} ⊆ {𝑦} → {𝑥𝜑} ∈ V)
96, 8sylbir 225 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
109exlimiv 1855 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
111, 10sylbi 207 1 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701  wcel 1987  ∃*wmo 2470  {cab 2607  Vcvv 3186  wss 3555  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by:  rmorabex  4889  euabex  4890
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