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Theorem iotaint 4880
 Description: Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
iotaint (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Proof of Theorem iotaint
StepHypRef Expression
1 iotauni 4879 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 uniintabim 3652 . 2 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
31, 2eqtrd 2072 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ∃!weu 1900  {cab 2026  ∪ cuni 3580  ∩ cint 3615  ℩cio 4865 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-iota 4867 This theorem is referenced by:  bdcriota  10003
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