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Theorem csbriotag 5402
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 2830 . . 3 (z = Az / x(y B φ) = A / x(y B φ))
2 dfsbcq2 2742 . . . 4 (z = A → ([z / x]φ[A / x]φ))
32riotabidv 5393 . . 3 (z = A → (y B [z / x]φ) = (y B [A / x]φ))
41, 3eqeq12d 2036 . 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 2536 . . 3 z V
6 nfs1v 1797 . . . 4 x[z / x]φ
7 nfcv 2160 . . . 4 xB
86, 7nfriota 5399 . . 3 x(y B [z / x]φ)
9 sbequ12 1636 . . . 4 (x = z → (φ ↔ [z / x]φ))
109riotabidv 5393 . . 3 (x = z → (y B φ) = (y B [z / x]φ))
115, 8, 10csbief 2866 . 2 z / x(y B φ) = (y B [z / x]φ)
124, 11vtoclg 2588 1 (A 𝑉A / x(y B φ) = (y B [A / x]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  [wsb 1627  [wsbc 2739  csb 2827  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-v 2535  df-sbc 2740  df-csb 2828  df-sn 3354  df-uni 3553  df-iota 4792  df-riota 5391
This theorem is referenced by: (None)
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