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Theorem csbsng 3422
Description: Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng (A 𝑉A / x{B} = {A / xB})

Proof of Theorem csbsng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3 (A 𝑉A / x{yy = B} = {y[A / x]y = B})
2 sbceq2g 2866 . . . 4 (A 𝑉 → ([A / x]y = By = A / xB))
32abbidv 2152 . . 3 (A 𝑉 → {y[A / x]y = B} = {yy = A / xB})
41, 3eqtrd 2069 . 2 (A 𝑉A / x{yy = B} = {yy = A / xB})
5 df-sn 3373 . . 3 {B} = {yy = B}
65csbeq2i 2870 . 2 A / x{B} = A / x{yy = B}
7 df-sn 3373 . 2 {A / xB} = {yy = A / xB}
84, 6, 73eqtr4g 2094 1 (A 𝑉A / x{B} = {A / xB})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {cab 2023  [wsbc 2758  csb 2846  {csn 3367
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-sn 3373
This theorem is referenced by: (None)
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