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Theorem rabeqsn 4185
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 2916 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4149 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2eqeq12i 2635 . 2 ({𝑥𝑉𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} = {𝑥𝑥 = 𝑋})
4 abbi 2734 . 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 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-rab 2916  df-sn 4149
This theorem is referenced by:  umgr2v2enb1  26308  k0004val0  37934
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