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Theorem absn 4386
Description: Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6065. (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 4369 . . 3 {𝑌} = {𝑥𝑥 = 𝑌}
21eqeq2i 2811 . 2 ({𝑥𝜑} = {𝑌} ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑌})
3 abbi 2914 . 2 (∀𝑥(𝜑𝑥 = 𝑌) ↔ {𝑥𝜑} = {𝑥𝑥 = 𝑌})
42, 3bitr4i 270 1 ({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1651   = wceq 1653  {cab 2785  {csn 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-sn 4369
This theorem is referenced by:  rabeqsn  4405  dfiota2  6065  dfaiota2  41935  aiotaval  41942
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