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Theorem iotaint 6310
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 6309 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 uniintab 4889 . . 3 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
32biimpi 219 . 2 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
41, 3eqtrd 2857 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ∃!weu 2652  {cab 2800   cuni 4813   cint 4851  cio 6291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-uni 4814  df-int 4852  df-iota 6293
This theorem is referenced by: (None)
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