Theorem csbriotag 5402

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 2831	. . 3 ⊢ (z = A → ⦋z / x⦌(℩y ∈ B φ) = ⦋A / x⦌(℩y ∈ B φ))
2		dfsbcq2 2743	. . . 4 ⊢ (z = A → ([z / x]φ ↔ [A / x]φ))
3	2	riotabidv 5393	. . 3 ⊢ (z = A → (℩y ∈ B [z / x]φ) = (℩y ∈ B [A / x]φ))
4	1, 3	eqeq12d 2037	. 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 2537	. . 3 ⊢ z ∈ V
6		nfs1v 1798	. . . 4 ⊢ Ⅎx[z / x]φ
7		nfcv 2161	. . . 4 ⊢ ℲxB
8	6, 7	nfriota 5399	. . 3 ⊢ Ⅎx(℩y ∈ B [z / x]φ)
9		sbequ12 1637	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
10	9	riotabidv 5393	. . 3 ⊢ (x = z → (℩y ∈ B φ) = (℩y ∈ B [z / x]φ))
11	5, 8, 10	csbief 2867	. 2 ⊢ ⦋z / x⦌(℩y ∈ B φ) = (℩y ∈ B [z / x]φ)
12	4, 11	vtoclg 2589	1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ))

Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1375  [wsb 1628  [wsbc 2740  ⦋csb 2828  ℩crio 5390

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-rex 2289  df-v 2536  df-sbc 2741  df-csb 2829  df-sn 3355  df-uni 3554  df-iota 4792  df-riota 5391

This theorem is referenced by: (None)