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Theorem csbvarg 3163
 Description: The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbvarg (A V[A / x]x = A)

Proof of Theorem csbvarg
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 vex 2862 . . . . . 6 y V
3 df-csb 3137 . . . . . . 7 [y / x]x = {z y / xz x}
4 sbcel2gv 3106 . . . . . . . 8 (y V → ([̣y / xz xz y))
54abbi1dv 2469 . . . . . . 7 (y V → {z y / xz x} = y)
63, 5syl5eq 2397 . . . . . 6 (y V → [y / x]x = y)
72, 6ax-mp 5 . . . . 5 [y / x]x = y
87csbeq2i 3162 . . . 4 [A / y][y / x]x = [A / y]y
9 csbco 3145 . . . 4 [A / y][y / x]x = [A / x]x
10 df-csb 3137 . . . 4 [A / y]y = {z A / yz y}
118, 9, 103eqtr3i 2381 . . 3 [A / x]x = {z A / yz y}
12 sbcel2gv 3106 . . . 4 (A V → ([̣A / yz yz A))
1312abbi1dv 2469 . . 3 (A V → {z A / yz y} = A)
1411, 13syl5eq 2397 . 2 (A V → [A / x]x = A)
151, 14syl 15 1 (A V[A / x]x = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  [̣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:  sbccsb2g  3165  csbfvg  5338
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