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Theorem sbcel1g 3155
 Description: Move proper substitution in and out of a membership relation. Note that the scope of [̣A / x]̣ is the wff B ∈ C, whereas the scope of [A / x] is the class B. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
sbcel1g (A V → ([̣A / xB C[A / x]B C))
Distinct variable group:   x,C
Allowed substitution hints:   A(x)   B(x)   V(x)

Proof of Theorem sbcel1g
StepHypRef Expression
1 sbcel12g 3151 . 2 (A V → ([̣A / xB C[A / x]B [A / x]C))
2 csbconstg 3150 . . 3 (A V[A / x]C = C)
32eleq2d 2420 . 2 (A V → ([A / x]B [A / x]C[A / x]B C))
41, 3bitrd 244 1 (A V → ([̣A / xB C[A / x]B C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by:  rspcsbela  3195
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