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Theorem sbccsb2g 3165
 Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
Assertion
Ref Expression
sbccsb2g (A V → ([̣A / xφA [A / x]{x φ}))

Proof of Theorem sbccsb2g
StepHypRef Expression
1 abid 2341 . . 3 (x {x φ} ↔ φ)
21sbcbii 3101 . 2 ([̣A / xx {x φ} ↔ [̣A / xφ)
3 sbcel12g 3151 . . 3 (A V → ([̣A / xx {x φ} ↔ [A / x]x [A / x]{x φ}))
4 csbvarg 3163 . . . 4 (A V[A / x]x = A)
54eleq1d 2419 . . 3 (A V → ([A / x]x [A / x]{x φ} ↔ A [A / x]{x φ}))
63, 5bitrd 244 . 2 (A V → ([̣A / xx {x φ} ↔ A [A / x]{x φ}))
72, 6syl5bbr 250 1 (A V → ([̣A / xφA [A / x]{x φ}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136 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  df-csb 3137 This theorem is referenced by: (None)
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