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Theorem iotaint 6075
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 6074 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 uniintab 4703 . . 3 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
32biimpi 208 . 2 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
41, 3eqtrd 2831 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  ∃!weu 2606  {cab 2783   cuni 4626   cint 4665  cio 6060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-sn 4367  df-pr 4369  df-uni 4627  df-int 4666  df-iota 6062
This theorem is referenced by: (None)
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