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Theorem sn-iotalemcor 40177
Description: Corollary of sn-iotalem 40176. Compare sb8iota 6398. (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 40176 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
21unieqi 4854 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
3 df-iota 6386 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
4 df-iota 6386 . 2 (℩𝑦{𝑥𝜑} = {𝑦}) = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
52, 3, 43eqtr4i 2776 1 (℩𝑥𝜑) = (℩𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {cab 2715  {csn 4563   cuni 4841  cio 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3433  df-in 3895  df-ss 3905  df-sn 4564  df-uni 4842  df-iota 6386
This theorem is referenced by: (None)
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