Theorem bj-csbsnlem 37400

Description: Lemma for bj-csbsn 37401 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-csbsnlem	⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-csbsnlem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abid 2747	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ [𝐴 / 𝑥]𝑦 ∈ {𝑥})
2		df-sbc 3748	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑥} ↔ 𝐴 ∈ {𝑥 ∣ 𝑦 ∈ {𝑥}})
3		clelab 2909	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑦 ∈ {𝑥}} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ {𝑥}))
4		velsn 4601	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
5	4	anbi2i 634	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 ∈ {𝑥}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝑥))
6	5	exbii 1871	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ {𝑥}) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝑥))
7		eqeq2 2777	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝑥 ↔ 𝑦 = 𝐴))
8	7	pm5.32i 584	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑥) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
9	8	exbii 1871	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
10		19.41v 1972	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝐴) ↔ (∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
11		simpr 489	. . . . . . 7 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
12		eqvisset 3477	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V)
13		elisset 2847	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 = 𝐴)
14	12, 13	syl 18	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
15	14	ancri 558	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
16	11, 15	impbii 212	. . . . . 6 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴) ↔ 𝑦 = 𝐴)
17	9, 10, 16	3bitri 300	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝑥) ↔ 𝑦 = 𝐴)
18	3, 6, 17	3bitri 300	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
19	1, 2, 18	3bitri 300	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
20		df-csb 3856	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}}
21	20	eleq2i 2857	. . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑥} ↔ 𝑦 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}})
22		velsn 4601	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
23	19, 21, 22	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑥} ↔ 𝑦 ∈ {𝐴})
24	23	eqriv 2762	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Syntax hints:   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  Vcvv 3457  [wsbc 3747  ⦋csb 3855  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-sbc 3748  df-csb 3856  df-sn 4586

This theorem is referenced by:  bj-csbsn  37401