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Theorem sbcel2 4022
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 4016 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3579 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
32eleq1d 2715 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝐶))
41, 3syl5bb 272 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
5 sbcex 3478 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
65con3i 150 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
7 noel 3952 . . . 4 ¬ 𝐵 ∈ ∅
8 csbprc 4013 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
98eleq2d 2716 . . . 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 176 1 ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 2030  Vcvv 3231  [wsbc 3468  csb 3566  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949
This theorem is referenced by:  csbcom  4027  sbccsb  4037  sbnfc2  4040  csbab  4041  sbcssg  4118  csbuni  4498  csbxp  5234  csbdm  5350  issubc  16542  esum2dlem  30282  bj-sbeq  33021  bj-sbceqgALT  33022  bj-sels  33075  f1omptsnlem  33313  csbcom2fi  34064  sbcssOLD  39073  csbabgOLD  39367  csbxpgOLD  39368  csbrngOLD  39371  disjinfi  39694  iccelpart  41694
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