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Theorem sbcel1g 3939
Description: Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵𝐶, whereas the scope of 𝐴 / 𝑥 is the class 𝐵. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
sbcel1g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcel1g
StepHypRef Expression
1 sbcel12 3935 . 2 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3512 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32eleq2d 2673 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐶))
41, 3syl5bb 271 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wcel 1977  [wsbc 3402  csb 3499
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:  rspcsbela  3958  csbopg  4353  fprodcllemf  14476  wunnat  16388  catcfuccl  16531  nbgraopALT  25747  rusgrasn  26266  esumpfinvalf  29259  esum2dlem  29275  measiuns  29401  bj-sbel1  31886  csbfinxpg  32195  finixpnum  32358  renegclALT  33061  cdlemk35s  35037  ellimcabssub0  38478
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