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Theorem rabeqsn 4161
 Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
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 2905 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4126 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2eqeq12i 2624 . 2 ({𝑥𝑉𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑥𝑥 = 𝑋})
4 abbi 2724 . 2 (∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋) ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑥𝑥 = 𝑋})
53, 4bitr4i 266 1 ({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905  df-sn 4126 This theorem is referenced by:  k0004val0  37472  umgr2v2enb1  40742
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