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Theorem sbcel2 4371
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 4364 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3906 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
32eleq1d 2902 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝐶))
41, 3syl5bb 284 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
5 sbcex 3786 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
65con3i 157 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
7 noel 4300 . . . 4 ¬ 𝐵 ∈ ∅
8 csbprc 4362 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
98eleq2d 2903 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐶𝐵 ∈ ∅))
107, 9mtbiri 328 . . 3 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝐶)
116, 102falsed 378 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
124, 11pm2.61i 183 1 ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wcel 2107  Vcvv 3500  [wsbc 3776  csb 3887  c0 4295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-nul 4296
This theorem is referenced by:  csbcom  4373  sbccsb  4389  sbnfc2  4392  csbab  4393  sbcssg  4466  csbuni  4865  csbxp  5649  csbdm  5765  issubc  17100  esum2dlem  31256  bj-sbeq  34121  bj-sbceqgALT  34122  f1omptsnlem  34505  csbcom2fi  35293  sbcssgVD  41101  csbingVD  41102  csbunigVD  41116  disjinfi  41338  iccelpart  43444
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