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Theorem sn-iotalem 41124
Description: An unused lemma showing that many equivalences involving df-iota 6495 are potentially provable without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
sn-iotalem {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotalem
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2736 . . . . . . 7 ({𝑥𝜑} = {𝑤} → ({𝑥𝜑} = {𝑧} ↔ {𝑤} = {𝑧}))
2 sneqbg 4844 . . . . . . . . 9 (𝑤 ∈ V → ({𝑤} = {𝑧} ↔ 𝑤 = 𝑧))
32elv 3480 . . . . . . . 8 ({𝑤} = {𝑧} ↔ 𝑤 = 𝑧)
4 equcom 2021 . . . . . . . 8 (𝑤 = 𝑧𝑧 = 𝑤)
53, 4bitri 274 . . . . . . 7 ({𝑤} = {𝑧} ↔ 𝑧 = 𝑤)
61, 5bitrdi 286 . . . . . 6 ({𝑥𝜑} = {𝑤} → ({𝑥𝜑} = {𝑧} ↔ 𝑧 = 𝑤))
7 sneq 4638 . . . . . . . . 9 (𝑦 = 𝑧 → {𝑦} = {𝑧})
87eqeq2d 2743 . . . . . . . 8 (𝑦 = 𝑧 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑧}))
98elabg 3666 . . . . . . 7 (𝑧 ∈ V → (𝑧 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ {𝑥𝜑} = {𝑧}))
109elv 3480 . . . . . 6 (𝑧 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ {𝑥𝜑} = {𝑧})
11 velsn 4644 . . . . . 6 (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
126, 10, 113bitr4g 313 . . . . 5 ({𝑥𝜑} = {𝑤} → (𝑧 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ 𝑧 ∈ {𝑤}))
1312eqrdv 2730 . . . 4 ({𝑥𝜑} = {𝑤} → {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤})
14 vsnid 4665 . . . . . 6 𝑤 ∈ {𝑤}
15 eleq2 2822 . . . . . 6 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤} → (𝑤 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ 𝑤 ∈ {𝑤}))
1614, 15mpbiri 257 . . . . 5 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤} → 𝑤 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}})
17 sneq 4638 . . . . . . . 8 (𝑦 = 𝑤 → {𝑦} = {𝑤})
1817eqeq2d 2743 . . . . . . 7 (𝑦 = 𝑤 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑤}))
1918elabg 3666 . . . . . 6 (𝑤 ∈ V → (𝑤 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ {𝑥𝜑} = {𝑤}))
2019elv 3480 . . . . 5 (𝑤 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ {𝑥𝜑} = {𝑤})
2116, 20sylib 217 . . . 4 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤} → {𝑥𝜑} = {𝑤})
2213, 21impbii 208 . . 3 ({𝑥𝜑} = {𝑤} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤})
23 sneq 4638 . . . . . 6 (𝑧 = 𝑤 → {𝑧} = {𝑤})
2423eqeq2d 2743 . . . . 5 (𝑧 = 𝑤 → ({𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤}))
2524elabg 3666 . . . 4 (𝑤 ∈ V → (𝑤 ∈ {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤}))
2625elv 3480 . . 3 (𝑤 ∈ {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤})
2722, 20, 263bitr4i 302 . 2 (𝑤 ∈ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ↔ 𝑤 ∈ {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}})
2827eqriv 2729 1 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  {cab 2709  Vcvv 3474  {csn 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sn 4629
This theorem is referenced by:  sn-iotalemcor  41125
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