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Theorem sbcel2gOLD 38581
 Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbcel2 3987 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcel2gOLD (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem sbcel2gOLD
StepHypRef Expression
1 sbcel12gOLD 38580 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2 csbconstg 3544 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
32eleq1d 2685 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝐶))
41, 3bitrd 268 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1989  [wsbc 3433  ⦋csb 3531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-sbc 3434  df-csb 3532 This theorem is referenced by:  sbcssOLD  38582  csbabgOLD  38876  csbunigOLD  38877  csbxpgOLD  38879  csbrngOLD  38882  sbcssgVD  38945  csbingVD  38946  csbunigVD  38960
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