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Theorem rabeqsn 4358
 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 3059 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4322 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2eqeq12i 2774 . 2 ({𝑥𝑉𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑥𝑥 = 𝑋})
4 abbi 2875 . 2 (∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋) ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑥𝑥 = 𝑋})
53, 4bitr4i 267 1 ({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383  ∀wal 1630   = wceq 1632   ∈ wcel 2139  {cab 2746  {crab 3054  {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 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-clab 2747  df-cleq 2753  df-clel 2756  df-rab 3059  df-sn 4322 This theorem is referenced by:  umgr2v2enb1  26653  clwwlknon1loop  27267  wlkl0  27549  rabeqsnd  29670  k0004val0  38972
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