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Theorem iotaval2 6531
Description: Version of iotaval 6534 using df-iota 6516 instead of dfiota2 6517. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
iotaval2 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotaval2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-iota 6516 . 2 (℩𝑥𝜑) = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
2 eqeq1 2739 . . . . 5 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} = {𝑤} ↔ {𝑦} = {𝑤}))
3 sneqbg 4848 . . . . . . 7 (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤))
43elv 3483 . . . . . 6 ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)
5 equcom 2015 . . . . . 6 (𝑦 = 𝑤𝑤 = 𝑦)
64, 5bitri 275 . . . . 5 ({𝑦} = {𝑤} ↔ 𝑤 = 𝑦)
72, 6bitrdi 287 . . . 4 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
87alrimiv 1925 . . 3 ({𝑥𝜑} = {𝑦} → ∀𝑤({𝑥𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
9 uniabio 6530 . . 3 (∀𝑤({𝑥𝜑} = {𝑤} ↔ 𝑤 = 𝑦) → {𝑤 ∣ {𝑥𝜑} = {𝑤}} = 𝑦)
108, 9syl 17 . 2 ({𝑥𝜑} = {𝑦} → {𝑤 ∣ {𝑥𝜑} = {𝑤}} = 𝑦)
111, 10eqtrid 2787 1 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  {cab 2712  Vcvv 3478  {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-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by:  iotauni2  6532  iotaval  6534  iotaex  6536
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