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Theorem rabeqsn 4569
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 3118 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
21eqeq1i 2806 . 2 ({𝑥𝑉𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑋})
3 absn 4546 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
42, 3bitri 278 1 ({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112  {cab 2779  {crab 3113  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-rab 3118  df-sn 4529
This theorem is referenced by:  rabsn  4620  umgr2v2enb1  27320  clwwlknon1loop  27887  wlkl0  28156  rabeqsnd  30275  zarclssn  31230  k0004val0  40854
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