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Theorem reuaiotaiota 41979
 Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
reuaiotaiota (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem reuaiotaiota
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsneu 41958 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
2 reuabaiotaiota 41978 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitri 267 1 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1656  ∃!weu 2639  {cab 2811  {csn 4399  ℩cio 6088  ℩'caiota 41974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-sn 4400  df-pr 4402  df-uni 4661  df-int 4700  df-iota 6090  df-aiota 41976 This theorem is referenced by:  aiotaexaiotaiota  41983  dfafv2  42028
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