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Theorem csbidmg 3128
Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
Assertion
Ref Expression
csbidmg  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem csbidmg
StepHypRef Expression
1 elex 2763 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 csbnest1g 3127 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ [_ A  /  x ]_ A  /  x ]_ B )
3 csbconstg 3086 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ A  =  A )
43csbeq1d 3079 . . 3  |-  ( A  e.  _V  ->  [_ [_ A  /  x ]_ A  /  x ]_ B  =  [_ A  /  x ]_ B
)
52, 4eqtrd 2222 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
61, 5syl 14 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   _Vcvv 2752   [_csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by: (None)
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