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Theorem iotain 39291
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 2589 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 vex 3353 . . . . 5 𝑦 ∈ V
32intsn 4669 . . . 4 {𝑦} = 𝑦
4 nfa1 2193 . . . . . . 7 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 2215 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
64, 5abbid 2883 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
7 df-sn 4335 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
86, 7syl6eqr 2817 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
98inteqd 4638 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
10 iotaval 6042 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
113, 9, 103eqtr4a 2825 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
1211exlimiv 2025 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
131, 12sylbi 208 1 (∃!𝑥𝜑 {𝑥𝜑} = (℩𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650   = wceq 1652  wex 1874  ∃!weu 2581  {cab 2751  {csn 4334   cint 4633  cio 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-sbc 3597  df-un 3737  df-in 3739  df-sn 4335  df-pr 4337  df-uni 4595  df-int 4634  df-iota 6031
This theorem is referenced by: (None)
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