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Theorem csbriotag 5423
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag (A 𝑉A / 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.
StepHypRef Expression
1 csbeq1 2849 . . 3 (z = Az / x(y B φ) = A / x(y B φ))
2 dfsbcq2 2761 . . . 4 (z = A → ([z / x]φ[A / x]φ))
32riotabidv 5413 . . 3 (z = A → (y B [z / x]φ) = (y B [A / x]φ))
41, 3eqeq12d 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 x[z / x]φ
7 nfcv 2175 . . . 4 xB
86, 7nfriota 5420 . . 3 x(y B [z / x]φ)
9 sbequ12 1651 . . . 4 (x = z → (φ ↔ [z / x]φ))
109riotabidv 5413 . . 3 (x = z → (y B φ) = (y B [z / x]φ))
115, 8, 10csbief 2885 . 2 z / x(y B φ) = (y B [z / x]φ)
124, 11vtoclg 2607 1 (A 𝑉A / x(y B φ) = (y B [A / x]φ))
Colors of variables: wff set class
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)
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