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Theorem csbid 3143
 Description: Analog of sbid 1922 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbid [x / x]A = A

Proof of Theorem csbid
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3137 . 2 [x / x]A = {y x / xy A}
2 sbsbc 3050 . . . 4 ([x / x]y A ↔ [̣x / xy A)
3 sbid 1922 . . . 4 ([x / x]y Ay A)
42, 3bitr3i 242 . . 3 ([̣x / xy Ay A)
54abbii 2465 . 2 {y x / xy A} = {y y A}
6 abid2 2470 . 2 {y y A} = A
71, 5, 63eqtri 2377 1 [x / x]A = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  [wsb 1648   ∈ 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbeq1a  3144  fvmpt2i  5703
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