Theorem bj-csbsnlem 37145

Description: Lemma for bj-csbsn 37146 (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 2719	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ [𝐴 / 𝑥]𝑦 ∈ {𝑥})
2		df-sbc 3743	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑥} ↔ 𝐴 ∈ {𝑥 ∣ 𝑦 ∈ {𝑥}})
3		clelab 2881	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑦 ∈ {𝑥}} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ {𝑥}))
4		velsn 4598	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
5	4	anbi2i 624	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 ∈ {𝑥}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝑥))
6	5	exbii 1850	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ {𝑥}) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝑥))
7		eqeq2 2749	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝑥 ↔ 𝑦 = 𝐴))
8	7	pm5.32i 574	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑥) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
9	8	exbii 1850	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
10		19.41v 1951	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝐴) ↔ (∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
11		simpr 484	. . . . . . 7 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
12		eqvisset 3462	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V)
13		elisset 2819	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 = 𝐴)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
15	14	ancri 549	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴))
16	11, 15	impbii 209	. . . . . 6 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ 𝑦 = 𝐴) ↔ 𝑦 = 𝐴)
17	9, 10, 16	3bitri 297	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑦 = 𝑥) ↔ 𝑦 = 𝐴)
18	3, 6, 17	3bitri 297	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
19	1, 2, 18	3bitri 297	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
20		df-csb 3852	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}}
21	20	eleq2i 2829	. . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑥} ↔ 𝑦 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ {𝑥}})
22		velsn 4598	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
23	19, 21, 22	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑥} ↔ 𝑦 ∈ {𝐴})
24	23	eqriv 2734	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  Vcvv 3442  [wsbc 3742  ⦋csb 3851  {csn 4582

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-sbc 3743  df-csb 3852  df-sn 4583

This theorem is referenced by:  bj-csbsn  37146