Theorem csbriotag 5423

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 2849	. . 3 ⊢ (z = A → ⦋z / x⦌(℩y ∈ B φ) = ⦋A / x⦌(℩y ∈ B φ))
2		dfsbcq2 2761	. . . 4 ⊢ (z = A → ([z / x]φ ↔ [A / x]φ))
3	2	riotabidv 5413	. . 3 ⊢ (z = A → (℩y ∈ B [z / x]φ) = (℩y ∈ B [A / x]φ))
4	1, 3	eqeq12d 2051	. 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 2554	. . 3 ⊢ z ∈ V
6		nfs1v 1812	. . . 4 ⊢ Ⅎx[z / x]φ
7		nfcv 2175	. . . 4 ⊢ ℲxB
8	6, 7	nfriota 5420	. . 3 ⊢ Ⅎx(℩y ∈ B [z / x]φ)
9		sbequ12 1651	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
10	9	riotabidv 5413	. . 3 ⊢ (x = z → (℩y ∈ B φ) = (℩y ∈ B [z / x]φ))
11	5, 8, 10	csbief 2885	. 2 ⊢ ⦋z / x⦌(℩y ∈ B φ) = (℩y ∈ B [z / x]φ)
12	4, 11	vtoclg 2607	1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ))

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  [wsb 1642  [wsbc 2758  ⦋csb 2846  ℩crio 5410

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-sn 3373  df-uni 3572  df-iota 4810  df-riota 5411

This theorem is referenced by: (None)