Theorem csbriota 7418

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 3854	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 3745	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 7405	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2779	. . 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 3458	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2196	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 7415	. . . 4 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 2290	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 7405	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3885	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 3521	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
13		csbprc 4370	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ∅)
14		df-riota 7403	. . . 4 ⊢ (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
15		euex 2605	. . . . . 6 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
16		sbcex 3752	. . . . . . . 8 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
17	16	adantl 487	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
18	17	exlimiv 1965	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
19	15, 18	syl 18	. . . . 5 ⊢ (∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
20		iotanul 6540	. . . . 5 ⊢ (¬ ∃!𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) = ∅)
21	19, 20	nsyl5 160	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑦(𝑦 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) = ∅)
22	14, 21	eqtr2id 2811	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
23	13, 22	eqtrd 2798	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
24	12, 23	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812  [wsb 2101   ∈ wcel 2147  ∃!weu 2596  Vcvv 3454  [wsbc 3742  ⦋csb 3851  ∅c0 4281  ℩cio 6513  ℩crio 7402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-nul 4282  df-sn 4589  df-uni 4871  df-iota 6515  df-riota 7403

This theorem is used by:  cdlemkid3N  42193  cdlemkid4  42194