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Theorem sn-iotalemcor 42261
Description: Corollary of sn-iotalem 42260. Compare sb8iota 6525. (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 42260 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
21unieqi 4919 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
3 df-iota 6514 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
4 df-iota 6514 . 2 (℩𝑦{𝑥𝜑} = {𝑦}) = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
52, 3, 43eqtr4i 2775 1 (℩𝑥𝜑) = (℩𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2714  {csn 4626   cuni 4907  cio 6512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-sn 4627  df-uni 4908  df-iota 6514
This theorem is referenced by: (None)
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