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Theorem absn 4568
Description: Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6303. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.)
Assertion
Ref Expression
absn ({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))
Distinct variable group:   𝑥,𝑌
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem absn
StepHypRef Expression
1 df-sn 4551 . . 3 {𝑌} = {𝑥𝑥 = 𝑌}
21eqeq2i 2837 . 2 ({𝑥𝜑} = {𝑌} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑌})
3 abbi 2891 . 2 (∀𝑥(𝜑𝑥 = 𝑌) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑌})
42, 3bitr4i 281 1 ({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  {cab 2802  {csn 4550
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 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-sn 4551
This theorem is referenced by:  rabeqsn  4591  euabsn2  4646  dfiota2  6303  dfaiota2  43569  aiotaval  43576
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