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Theorem sbc6g 3071
 Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
sbc6g (A V → ([̣A / xφx(x = Aφ)))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem sbc6g
StepHypRef Expression
1 nfe1 1732 . . 3 xx(x = A φ)
2 ceqex 2969 . . 3 (x = A → (φx(x = A φ)))
31, 2ceqsalg 2883 . 2 (A V → (x(x = Aφ) ↔ x(x = A φ)))
4 sbc5 3070 . 2 ([̣A / xφx(x = A φ))
53, 4syl6rbbr 255 1 (A V → ([̣A / xφx(x = Aφ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ 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:  sbc6  3072  sbciegft  3076  ralsns  3763
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