Theorem csbiota 6524

Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbiota	⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

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

Proof of Theorem csbiota

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3877	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑))
2		dfsbcq2 3768	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	iotabidv 6515	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2751	. . 3 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5		vex 3463	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2156	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	6	nfiotaw 6488	. . . 4 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8		sbequ12 2251	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	8	iotabidv 6515	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
10	5, 7, 9	csbief 3908	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
11	4, 10	vtoclg 3533	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12		csbprc 4384	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = ∅)
13		sbcex 3775	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1936	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16		euex 2576	. . . . 5 ⊢ (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
17	16	con3i 154	. . . 4 ⊢ (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18		iotanul 6509	. . . 4 ⊢ (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
19	15, 17, 18	3syl 18	. . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
20	12, 19	eqtr4d 2773	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
21	11, 20	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   = wceq 1540  ∃wex 1779  [wsb 2064   ∈ wcel 2108  ∃!weu 2567  Vcvv 3459  [wsbc 3765  ⦋csb 3874  ∅c0 4308  ℩cio 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-ss 3943  df-nul 4309  df-sn 4602  df-uni 4884  df-iota 6484

This theorem is referenced by:  csbfv12  6924