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Theorem sn-iotalemcor 42240
Description: Corollary of sn-iotalem 42239. Compare sb8iota 6527. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
sn-iotalemcor (℩𝑥𝜑) = (℩𝑦{𝑥𝜑} = {𝑦})
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotalemcor
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sn-iotalem 42239 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
21unieqi 4924 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
3 df-iota 6516 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
4 df-iota 6516 . 2 (℩𝑦{𝑥𝜑} = {𝑦}) = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
52, 3, 43eqtr4i 2773 1 (℩𝑥𝜑) = (℩𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {cab 2712  {csn 4631   cuni 4912  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-sn 4632  df-uni 4913  df-iota 6516
This theorem is referenced by: (None)
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