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Theorem moabex 5076
 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 2614 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abss 3812 . . . . 5 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4337 . . . . . . 7 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43imbi2i 325 . . . . . 6 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1896 . . . . 5 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 264 . . . 4 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
7 snex 5057 . . . . 5 {𝑦} ∈ V
87ssex 4954 . . . 4 ({𝑥𝜑} ⊆ {𝑦} → {𝑥𝜑} ∈ V)
96, 8sylbir 225 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
109exlimiv 2007 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
111, 10sylbi 207 1 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1630  ∃wex 1853   ∈ wcel 2139  ∃*wmo 2608  {cab 2746  Vcvv 3340   ⊆ wss 3715  {csn 4321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324 This theorem is referenced by:  rmorabex  5077  euabex  5078
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