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Theorem iotaint 6021
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 6020 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 uniintab 4663 . . 3 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
32biimpi 206 . 2 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
41, 3eqtrd 2790 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1628  ∃!weu 2603  {cab 2742   cuni 4584   cint 4623  cio 6006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-sn 4318  df-pr 4320  df-uni 4585  df-int 4624  df-iota 6008
This theorem is referenced by: (None)
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