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Theorem iotaint 5166
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 5165 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 uniintabim 3861 . 2 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
31, 2eqtrd 2198 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1343  ∃!weu 2014  {cab 2151   cuni 3789   cint 3824  cio 5151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-iota 5153
This theorem is referenced by:  bdcriota  13765
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