Theorem csbriotag 6025

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 3144	. . 3 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 3048	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 6013	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2249	. 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 2818	. . 3 ⊢ 𝑧 ∈ V
6		nfs1v 1995	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2386	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 6021	. . 3 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 1820	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 6013	. . 3 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3186	. 2 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 2877	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1398  [wsb 1811   ∈ wcel 2205  [wsbc 3045  ⦋csb 3141  ℩crio 6010

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-sn 3700  df-uni 3920  df-iota 5317  df-riota 6011

This theorem is referenced by: (None)