Theorem csbiota 6530

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 3853	. . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑))
2		dfsbcq2 3745	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	iotabidv 6521	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2778	. . 3 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5		vex 3457	. . . 4 ⊢ 𝑧 ∈ V
6		nfs1v 2193	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	6	nfiotaw 6497	. . . 4 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8		sbequ12 2288	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	8	iotabidv 6521	. . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
10	5, 7, 9	csbief 3884	. . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
11	4, 10	vtoclg 3520	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
12		csbprc 4370	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = ∅)
13		sbcex 3752	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
14	13	con3i 155	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
15	14	nexdv 1969	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑)
16		euex 2604	. . . . 5 ⊢ (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑)
17	16	con3i 155	. . . 4 ⊢ (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑)
18		iotanul 6517	. . . 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 2800	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
21	11, 20	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∃wex 1812  [wsb 2099   ∈ wcel 2145  ∃!weu 2595  Vcvv 3453  [wsbc 3742  ⦋csb 3850  ∅c0 4282  ℩cio 6491

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-ss 3919  df-nul 4283  df-sn 4588  df-uni 4871  df-iota 6493

This theorem is used by:  csbfv12  6927