Theorem csbriota 7420

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 3924	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 3807	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 7406	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2756	. . 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 3492	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2157	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 7417	. . . 4 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 2252	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 7406	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3956	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 3566	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
13		csbprc 4432	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ∅)
14		df-riota 7404	. . . 4 ⊢ (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
15		euex 2580	. . . . . 6 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
16		sbcex 3814	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
18	17	exlimiv 1929	. . . . . 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 6551	. . . . 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 2793	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
23	13, 22	eqtrd 2780	. 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 395   = wceq 1537  ∃wex 1777  [wsb 2064   ∈ wcel 2108  ∃!weu 2571  Vcvv 3488  [wsbc 3804  ⦋csb 3921  ∅c0 4352  ℩cio 6523  ℩crio 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649  df-uni 4932  df-iota 6525  df-riota 7404

This theorem is referenced by:  cdlemkid3N  40890  cdlemkid4  40891