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Theorem sbcel1g 4131
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 4127 . 2 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3695 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32eleq2d 2836 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐶))
41, 3syl5bb 272 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  [wsbc 3587  csb 3682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-nul 4064
This theorem is referenced by:  rspcsbela  4150  csbopg  4557  fprodcllemf  14891  wunnat  16819  catcfuccl  16962  esumpfinvalf  30474  esum2dlem  30490  measiuns  30616  bj-sbel1  33225  csbfinxpg  33558  finixpnum  33723  renegclALT  34767  cdlemk35s  36743  ellimcabssub0  40364
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