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Theorem iotauni 4351
 Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iotauni (∃!xφ → (℩xφ) = {x φ})

Proof of Theorem iotauni
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2208 . 2 (∃!xφzx(φx = z))
2 iotaval 4350 . . . 4 (x(φx = z) → (℩xφ) = z)
3 uniabio 4349 . . . 4 (x(φx = z) → {x φ} = z)
42, 3eqtr4d 2388 . . 3 (x(φx = z) → (℩xφ) = {x φ})
54exlimiv 1634 . 2 (zx(φx = z) → (℩xφ) = {x φ})
61, 5sylbi 187 1 (∃!xφ → (℩xφ) = {x φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  ∪cuni 3891  ℩cio 4337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  iotaint  4352  iotassuni  4355  dfiota3  4370  dfiota4  4372
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