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Theorem csbsng 3785
 Description: Distribute proper substitution through the singleton of a class. csbsng 3785 is derived from the virtual deduction proof csbsngVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng (A V[A / x]{B} = {[A / x]B})

Proof of Theorem csbsng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbabg 3197 . . 3 (A V[A / x]{y y = B} = {y A / xy = B})
2 sbceq2g 3158 . . . 4 (A V → ([̣A / xy = By = [A / x]B))
32abbidv 2467 . . 3 (A V → {y A / xy = B} = {y y = [A / x]B})
41, 3eqtrd 2385 . 2 (A V[A / x]{y y = B} = {y y = [A / x]B})
5 df-sn 3741 . . 3 {B} = {y y = B}
65csbeq2i 3162 . 2 [A / x]{B} = [A / x]{y y = B}
7 df-sn 3741 . 2 {[A / x]B} = {y y = [A / x]B}
84, 6, 73eqtr4g 2410 1 (A V[A / x]{B} = {[A / x]B})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136  {csn 3737 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  df-sn 3741 This theorem is referenced by: (None)
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