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Theorem iotaint 5781
 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 5780 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 uniintab 4450 . . 3 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
32biimpi 205 . 2 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
41, 3eqtrd 2644 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∃!weu 2458  {cab 2596  ∪ cuni 4372  ∩ cint 4410  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411  df-iota 5768 This theorem is referenced by: (None)
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