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Theorem csbidmg 3135
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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 csbnest1g 3133 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ [_ A  /  x ]_ A  /  x ]_ B )
3 csbconstg 3095 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ A  =  A )
43csbeq1d 3087 . . 3  |-  ( A  e.  _V  ->  [_ [_ A  /  x ]_ A  /  x ]_ B  =  [_ A  /  x ]_ B
)
52, 4eqtrd 2315 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
61, 5syl 15 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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