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Theorem csbidmg 2985
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 2631 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 csbnest1g 2984 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ [_ A  /  x ]_ A  /  x ]_ B )
3 csbconstg 2946 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ A  =  A )
43csbeq1d 2940 . . 3  |-  ( A  e.  _V  ->  [_ [_ A  /  x ]_ A  /  x ]_ B  =  [_ A  /  x ]_ B
)
52, 4eqtrd 2121 . 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 1290    e. wcel 1439   _Vcvv 2620   [_csb 2934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842  df-csb 2935
This theorem is referenced by: (None)
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