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Theorem iotain 41708
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 2573 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 vex 3412 . . . . 5 𝑦 ∈ V
32intsn 4897 . . . 4 {𝑦} = 𝑦
4 abbi1 2806 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
5 df-sn 4542 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
64, 5eqtr4di 2796 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
76inteqd 4864 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
8 iotaval 6354 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
93, 7, 83eqtr4a 2804 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
109exlimiv 1938 . 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 1541   = wceq 1543  wex 1787  ∃!weu 2567  {cab 2714  {csn 4541   cint 4859  cio 6336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-v 3410  df-un 3871  df-in 3873  df-ss 3883  df-sn 4542  df-pr 4544  df-uni 4820  df-int 4860  df-iota 6338
This theorem is referenced by: (None)
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