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Theorem sbcel2gv 3106
 Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel2gv (B V → ([̣B / xA xA B))
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   V(x)

Proof of Theorem sbcel2gv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . 2 (y = B → ([y / x]A x ↔ [̣B / xA x))
2 eleq2 2414 . 2 (y = B → (A yA B))
3 nfv 1619 . . 3 x A y
4 eleq2 2414 . . 3 (x = y → (A xA y))
53, 4sbie 2038 . 2 ([y / x]A xA y)
61, 2, 5vtoclbg 2915 1 (B V → ([̣B / xA xA B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem is referenced by:  csbvarg  3163
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