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Theorem sbcel1gv 3105
 Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel1gv (A V → ([̣A / xx BA B))
Distinct variable group:   x,B
Allowed substitution hints:   A(x)   V(x)

Proof of Theorem sbcel1gv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . 2 (y = A → ([y / x]x B ↔ [̣A / xx B))
2 eleq1 2413 . 2 (y = A → (y BA B))
3 clelsb3 2455 . 2 ([y / x]x By B)
41, 2, 3vtoclbg 2915 1 (A V → ([̣A / xx BA 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-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 This theorem is referenced by: (None)
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