Theorem csbriotag 5392

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 2823	. . 3 ⊢ (z = A → ⦋z / x⦌(℩y ∈ B φ) = ⦋A / x⦌(℩y ∈ B φ))
2		dfsbcq2 2735	. . . 4 ⊢ (z = A → ([z / x]φ ↔ [A / x]φ))
3	2	riotabidv 5383	. . 3 ⊢ (z = A → (℩y ∈ B [z / x]φ) = (℩y ∈ B [A / x]φ))
4	1, 3	eqeq12d 2027	. 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 2529	. . 3 ⊢ z ∈ V
6		nfs1v 1788	. . . 4 ⊢ Ⅎx[z / x]φ
7		nfcv 2151	. . . 4 ⊢ ℲxB
8	6, 7	nfriota 5389	. . 3 ⊢ Ⅎx(℩y ∈ B [z / x]φ)
9		sbequ12 1627	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
10	9	riotabidv 5383	. . 3 ⊢ (x = z → (℩y ∈ B φ) = (℩y ∈ B [z / x]φ))
11	5, 8, 10	csbief 2859	. 2 ⊢ ⦋z / x⦌(℩y ∈ B φ) = (℩y ∈ B [z / x]φ)
12	4, 11	vtoclg 2581	1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ))

Syntax hints:   → wi 4   = wceq 1223   ∈ wcel 1366  [wsb 1618  [wsbc 2732  ⦋csb 2820  ℩crio 5380

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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995

This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-v 2528  df-sbc 2733  df-csb 2821  df-sn 3345  df-uni 3544  df-iota 4782  df-riota 5381

This theorem is referenced by: (None)