Theorem csbriota 7248

Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)

Assertion

Ref	Expression
csbriota	⊢ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem csbriota

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3835	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 3719	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 7234	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2754	. . 3 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)))
5		vex 3436	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2153	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 7245	. . . 4 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 2244	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 7234	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3867	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 3505	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
13		csbprc 4340	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ∅)
14		df-riota 7232	. . . 4 ⊢ (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
15		euex 2577	. . . . . 6 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
16		sbcex 3726	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
17	16	adantl 482	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
18	17	exlimiv 1933	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
19	15, 18	syl 17	. . . . 5 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
20		iotanul 6411	. . . . 5 ⊢ (¬ ∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) = ∅)
21	19, 20	nsyl5 159	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) = ∅)
22	14, 21	eqtr2id 2791	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
23	13, 22	eqtrd 2778	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
24	12, 23	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1539  ∃wex 1782  [wsb 2067   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432  [wsbc 3716  ⦋csb 3832  ∅c0 4256  ℩cio 6389  ℩crio 7231

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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-uni 4840  df-iota 6391  df-riota 7232

This theorem is referenced by:  cdlemkid3N  38947  cdlemkid4  38948