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Theorem sbcel2 3941
 Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcel2 ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem sbcel2
StepHypRef Expression
1 sbcel12 3935 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3512 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
32eleq1d 2672 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝐶))
41, 3syl5bb 271 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
5 sbcex 3412 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
65con3i 149 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
7 noel 3878 . . . 4 ¬ 𝐵 ∈ ∅
8 csbprc 3932 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
98eleq2d 2673 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐶𝐵 ∈ ∅))
107, 9mtbiri 316 . . 3 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝐶)
116, 102falsed 365 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
124, 11pm2.61i 175 1 ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875 This theorem is referenced by:  csbcom  3946  sbccsb  3956  sbnfc2  3959  csbab  3960  sbcssg  4035  csbuni  4402  csbxp  5123  csbdm  5240  issubc  16318  nbgraopALT  25953  esum2dlem  29481  bj-sbeq  32088  bj-sbceqgALT  32089  bj-sels  32143  f1omptsnlem  32359  csbcom2fi  33104  disjinfi  38375  iccelpart  39971
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