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Theorem iotain 38438
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 df-eu 2472 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 vex 3198 . . . . 5 𝑦 ∈ V
32intsn 4504 . . . 4 {𝑦} = 𝑦
4 nfa1 2026 . . . . . . 7 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 2051 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
64, 5abbid 2738 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
7 df-sn 4169 . . . . . 6 {𝑦} = {𝑥𝑥 = 𝑦}
86, 7syl6eqr 2672 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
98inteqd 4471 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
10 iotaval 5850 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
113, 9, 103eqtr4a 2680 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
1211exlimiv 1856 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = (℩𝑥𝜑))
131, 12sylbi 207 1 (∃!𝑥𝜑 {𝑥𝜑} = (℩𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479   = wceq 1481  wex 1702  ∃!weu 2468  {cab 2606  {csn 4168   cint 4466  cio 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-sbc 3430  df-un 3572  df-in 3574  df-sn 4169  df-pr 4171  df-uni 4428  df-int 4467  df-iota 5839
This theorem is referenced by: (None)
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