Theorem sn-iotalem 40189

Description: An unused lemma showing that many equivalences involving df-iota 6391 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 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {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.

Step	Hyp	Ref	Expression
1		eqeq1 2742	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑤} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑤} = {𝑧}))
2		sneqbg 4774	. . . . . . . . 9 ⊢ (𝑤 ∈ V → ({𝑤} = {𝑧} ↔ 𝑤 = 𝑧))
3	2	elv 3438	. . . . . . . 8 ⊢ ({𝑤} = {𝑧} ↔ 𝑤 = 𝑧)
4		equcom 2021	. . . . . . . 8 ⊢ (𝑤 = 𝑧 ↔ 𝑧 = 𝑤)
5	3, 4	bitri 274	. . . . . . 7 ⊢ ({𝑤} = {𝑧} ↔ 𝑧 = 𝑤)
6	1, 5	bitrdi 287	. . . . . 6 ⊢ ({𝑥 ∣ 𝜑} = {𝑤} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ 𝑧 = 𝑤))
7		sneq 4571	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
8	7	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑧}))
9	8	elabg 3607	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ {𝑥 ∣ 𝜑} = {𝑧}))
10	9	elv 3438	. . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ {𝑥 ∣ 𝜑} = {𝑧})
11		velsn 4577	. . . . . 6 ⊢ (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
12	6, 10, 11	3bitr4g 314	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑤} → (𝑧 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ 𝑧 ∈ {𝑤}))
13	12	eqrdv 2736	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑤} → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤})
14		vsnid 4598	. . . . . 6 ⊢ 𝑤 ∈ {𝑤}
15		eleq2 2827	. . . . . 6 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤} → (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ 𝑤 ∈ {𝑤}))
16	14, 15	mpbiri 257	. . . . 5 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤} → 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}})
17		sneq 4571	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
18	17	eqeq2d 2749	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤}))
19	18	elabg 3607	. . . . . 6 ⊢ (𝑤 ∈ V → (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ {𝑥 ∣ 𝜑} = {𝑤}))
20	19	elv 3438	. . . . 5 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ {𝑥 ∣ 𝜑} = {𝑤})
21	16, 20	sylib 217	. . . 4 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤} → {𝑥 ∣ 𝜑} = {𝑤})
22	13, 21	impbii 208	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤})
23		sneq 4571	. . . . . 6 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
24	23	eqeq2d 2749	. . . . 5 ⊢ (𝑧 = 𝑤 → ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧} ↔ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤}))
25	24	elabg 3607	. . . 4 ⊢ (𝑤 ∈ V → (𝑤 ∈ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} ↔ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤}))
26	25	elv 3438	. . 3 ⊢ (𝑤 ∈ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} ↔ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤})
27	22, 20, 26	3bitr4i 303	. 2 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ↔ 𝑤 ∈ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}})
28	27	eqriv 2735	1 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3432  {csn 4561

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 3434  df-sn 4562

This theorem is referenced by:  sn-iotalemcor  40190