Theorem csbriotag 5886

Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)

Assertion

Ref	Expression
csbriotag	⊢ (𝐴 ∈ 𝑉 → ⦋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)   𝑉(𝑥,𝑦)

Proof of Theorem csbriotag

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3083	. . 3 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 2988	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 5875	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2208	. 2 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)))
5		vex 2763	. . 3 ⊢ 𝑧 ∈ V
6		nfs1v 1955	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2336	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 5883	. . 3 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 1782	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 5875	. . 3 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3125	. 2 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 2820	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1364  [wsb 1773   ∈ wcel 2164  [wsbc 2985  ⦋csb 3080  ℩crio 5872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-sn 3624  df-uni 3836  df-iota 5215  df-riota 5873

This theorem is referenced by: (None)