Theorem csbriota 6663

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 3569	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 3471	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 6653	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2666	. . 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 3234	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2465	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2793	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 6660	. . . 4 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 2149	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 6653	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3591	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 3297	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
13		csbprc 4013	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ∅)
14		df-riota 6651	. . . 4 ⊢ (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
15		euex 2522	. . . . . . 7 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
16		sbcex 3478	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
18	17	exlimiv 1898	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
19	15, 18	syl 17	. . . . . 6 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
20	19	con3i 150	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
21		iotanul 5904	. . . . 5 ⊢ (¬ ∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) = ∅)
22	20, 21	syl 17	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) = ∅)
23	14, 22	syl5req 2698	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
24	13, 23	eqtrd 2685	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
25	12, 24	pm2.61i 176	1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1523  ∃wex 1744  [wsb 1937   ∈ wcel 2030  ∃!weu 2498  Vcvv 3231  [wsbc 3468  ⦋csb 3566  ∅c0 3948  ℩cio 5887  ℩crio 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469  df-iota 5889  df-riota 6651

This theorem is referenced by:  cdlemkid3N  36538  cdlemkid4  36539