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Theorem csbdmg 4472
Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
csbdmg (A 𝑉A / xdom B = dom A / xB)

Proof of Theorem csbdmg
Dummy variables w y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3 (A 𝑉A / x{ywy, w B} = {y[A / x]wy, w B})
2 sbcex2 2806 . . . . 5 ([A / x]wy, w Bw[A / x]y, w B)
3 sbcel2g 2865 . . . . . 6 (A 𝑉 → ([A / x]y, w B ↔ ⟨y, w A / xB))
43exbidv 1703 . . . . 5 (A 𝑉 → (w[A / x]y, w Bwy, w A / xB))
52, 4syl5bb 181 . . . 4 (A 𝑉 → ([A / x]wy, w Bwy, w A / xB))
65abbidv 2152 . . 3 (A 𝑉 → {y[A / x]wy, w B} = {ywy, w A / xB})
71, 6eqtrd 2069 . 2 (A 𝑉A / x{ywy, w B} = {ywy, w A / xB})
8 dfdm3 4465 . . 3 dom B = {ywy, w B}
98csbeq2i 2870 . 2 A / xdom B = A / x{ywy, w B}
10 dfdm3 4465 . 2 dom A / xB = {ywy, w A / xB}
117, 9, 103eqtr4g 2094 1 (A 𝑉A / xdom B = dom A / xB)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wex 1378   wcel 1390  {cab 2023  [wsbc 2758  csb 2846  cop 3370  dom cdm 4288
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-br 3756  df-dm 4298
This theorem is referenced by:  sbcfng  4987
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