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Theorem sbcel2 4424
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 4417 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3927 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
32eleq1d 2824 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝐶))
41, 3bitrid 283 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
5 sbcex 3801 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
65con3i 154 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
7 noel 4344 . . . 4 ¬ 𝐵 ∈ ∅
8 csbprc 4415 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
98eleq2d 2825 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐶𝐵 ∈ ∅))
107, 9mtbiri 327 . . 3 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝐶)
116, 102falsed 376 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
124, 11pm2.61i 182 1 ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wcel 2106  Vcvv 3478  [wsbc 3791  csb 3908  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340
This theorem is referenced by:  csbcom  4426  sbccsb  4442  sbnfc2  4445  csbab  4446  sbcssg  4526  csbuni  4941  csbxp  5788  csbdm  5911  issubc  17886  esum2dlem  34073  weiunlem2  36446  bj-sbeq  36884  bj-sbceqgALT  36885  f1omptsnlem  37319  csbcom2fi  38115  sbcssgVD  44881  csbingVD  44882  csbunigVD  44896  disjinfi  45135  iccelpart  47358
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