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Theorem iotain 41136
 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 2634 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 vex 3444 . . . . 5 𝑦 ∈ V
32intsn 4874 . . . 4 {𝑦} = 𝑦
4 abbi1 2861 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
5 df-sn 4526 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
64, 5eqtr4di 2851 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
76inteqd 4843 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
8 iotaval 6298 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
93, 7, 83eqtr4a 2859 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
109exlimiv 1931 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
111, 10sylbi 220 1 (∃!𝑥𝜑 {𝑥𝜑} = (℩𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃wex 1781  ∃!weu 2628  {cab 2776  {csn 4525  ∩ cint 4838  ℩cio 6281 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801  df-int 4839  df-iota 6283 This theorem is referenced by: (None)
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