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Theorem sbcel2g 2865
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
sbcel2g (A 𝑉 → ([A / x]B 𝐶B A / x𝐶))
Distinct variable group:   x,B
Allowed substitution hints:   A(x)   𝐶(x)   𝑉(x)

Proof of Theorem sbcel2g
StepHypRef Expression
1 sbcel12g 2859 . 2 (A 𝑉 → ([A / x]B 𝐶A / xB A / x𝐶))
2 csbconstg 2858 . . 3 (A 𝑉A / xB = B)
32eleq1d 2103 . 2 (A 𝑉 → (A / xB A / x𝐶B A / x𝐶))
41, 3bitrd 177 1 (A 𝑉 → ([A / x]B 𝐶B A / x𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  [wsbc 2758  csb 2846
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
This theorem is referenced by:  csbcomg  2867  sbccsbg  2872  sbnfc2  2900  csbabg  2901  sbcssg  3324  csbunig  3579  csbxpg  4364  csbdmg  4472  csbrng  4725  bj-sels  9345
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