Theorem csbiota 6518

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 3858	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑))
2		dfsbcq2 3750	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	iotabidv 6509	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2781	. . 3 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5		vex 3461	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2193	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	6	nfiotaw 6485	. . . 4 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8		sbequ12 2289	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	8	iotabidv 6509	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
10	5, 7, 9	csbief 3889	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
11	4, 10	vtoclg 3525	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12		csbprc 4366	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = ∅)
13		sbcex 3757	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 155	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1959	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16		euex 2607	. . . . 5 ⊢ (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
17	16	con3i 155	. . . 4 ⊢ (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18		iotanul 6505	. . . 4 ⊢ (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
19	15, 17, 18	3syl 19	. . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅)
20	12, 19	eqtr4d 2803	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
21	11, 20	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   = wceq 1563  ∃wex 1802  [wsb 2093   ∈ wcel 2145  ∃!weu 2598  Vcvv 3457  [wsbc 3747  ⦋csb 3855  ∅c0 4288  ℩cio 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-ss 3924  df-nul 4289  df-sn 4586  df-uni 4868  df-iota 6481

This theorem is referenced by:  csbfv12  6916