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Theorem iotain 42789
Description: Equivalence between two different forms of . (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
iotain (∃!𝑥𝜑 {𝑥𝜑} = (℩𝑥𝜑))

Proof of Theorem iotain
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eu6 2569 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 vex 3451 . . . . 5 𝑦 ∈ V
32intsn 4951 . . . 4 {𝑦} = 𝑦
4 abbi 2801 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
5 df-sn 4591 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
64, 5eqtr4di 2791 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
76inteqd 4916 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
8 iotaval 6471 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
93, 7, 83eqtr4a 2799 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
109exlimiv 1934 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
111, 10sylbi 216 1 (∃!𝑥𝜑 {𝑥𝜑} = (℩𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  wex 1782  ∃!weu 2563  {cab 2710  {csn 4590   cint 4911  cio 6450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593  df-uni 4870  df-int 4912  df-iota 6452
This theorem is referenced by: (None)
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