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Theorem sbcel1g 4414
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 4409 . 2 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3913 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32eleq2d 2820 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐶))
41, 3bitrid 283 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  [wsbc 3778  csb 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-nul 4324
This theorem is referenced by:  rspcsbela  4436  csbopg  4892  fprodcllemf  15902  wunnat  17907  wunnatOLD  17908  catcfuccl  18069  catcfucclOLD  18070  esumpfinvalf  33074  esum2dlem  33090  measiuns  33215  bj-sbel1  35785  csbfinxpg  36269  finixpnum  36473  renegclALT  37833  cdlemk35s  39808  ellimcabssub0  44333
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