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Theorem absn 4575
Description: Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6308. (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 4558 . . 3 {𝑌} = {𝑥𝑥 = 𝑌}
21eqeq2i 2831 . 2 ({𝑥𝜑} = {𝑌} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑌})
3 abbi 2885 . 2 (∀𝑥(𝜑𝑥 = 𝑌) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑌})
42, 3bitr4i 279 1 ({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1526   = wceq 1528  {cab 2796  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-sn 4558
This theorem is referenced by:  rabeqsn  4596  euabsn2  4653  dfiota2  6308  dfaiota2  43163  aiotaval  43170
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