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Theorem sbcel1g 4422
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 4417 . 2 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2 csbconstg 3927 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32eleq2d 2825 . 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 206  wcel 2106  [wsbc 3791  csb 3908
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:  rspcsbela  4444  csbopg  4896  fprodcllemf  15991  wunnat  18011  wunnatOLD  18012  catcfuccl  18173  catcfucclOLD  18174  esumpfinvalf  34057  esum2dlem  34073  measiuns  34198  bj-sbel1  36888  csbfinxpg  37371  finixpnum  37592  renegclALT  38945  cdlemk35s  40920  ellimcabssub0  45573
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