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Theorem sbcel2 3966
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 3960 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3531 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
32eleq1d 2683 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝐶))
41, 3syl5bb 272 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
5 sbcex 3431 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
65con3i 150 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
7 noel 3900 . . . 4 ¬ 𝐵 ∈ ∅
8 csbprc 3957 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
98eleq2d 2684 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐶𝐵 ∈ ∅))
107, 9mtbiri 317 . . 3 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝐶)
116, 102falsed 366 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
124, 11pm2.61i 176 1 ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 1987  Vcvv 3189  [wsbc 3421  csb 3518  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-nul 3897
This theorem is referenced by:  csbcom  3971  sbccsb  3981  sbnfc2  3984  csbab  3985  sbcssg  4062  csbuni  4437  csbxp  5166  csbdm  5283  issubc  16427  esum2dlem  29959  bj-sbeq  32578  bj-sbceqgALT  32579  bj-sels  32632  f1omptsnlem  32850  csbcom2fi  33601  disjinfi  38885  iccelpart  40693
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