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Theorem rabeqsn 4668
Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 26-Aug-2022.)
Assertion
Ref Expression
rabeqsn ({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
Distinct variable group:   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabeqsn
StepHypRef Expression
1 df-rab 3431 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
21eqeq1i 2735 . 2 ({𝑥𝑉𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑋})
3 absn 4645 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
42, 3bitri 274 1 ({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wal 1537   = wceq 1539  wcel 2104  {cab 2707  {crab 3430  {csn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-rab 3431  df-sn 4628
This theorem is referenced by:  rabeqsnd  4670  rabsn  4724  made0  27605  umgr2v2enb1  29050  clwwlknon1loop  29618  wlkl0  29887  zarclssn  33151  k0004val0  43207
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