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Theorem iotauni 6328
Description: Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iotauni (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Proof of Theorem iotauni
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eu6 2657 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 iotaval 6327 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3 uniabio 6326 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → {𝑥𝜑} = 𝑧)
42, 3eqtr4d 2864 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
54exlimiv 1924 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
61, 5sylbi 218 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528   = wceq 1530  wex 1773  ∃!weu 2651  {cab 2804   cuni 4837  cio 6310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rex 3149  df-v 3502  df-sbc 3777  df-un 3945  df-sn 4565  df-pr 4567  df-uni 4838  df-iota 6312
This theorem is referenced by:  iotaint  6329  iotassuni  6332  dfiota4  6344  fveu  6658  riotauni  7112  afv2eu  43303
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