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Theorem csbidmg 3221
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 2881 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 csbnest1g 3219 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ [_ A  /  x ]_ A  /  x ]_ B )
3 csbconstg 3181 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ A  =  A )
43csbeq1d 3173 . . 3  |-  ( A  e.  _V  ->  [_ [_ A  /  x ]_ A  /  x ]_ B  =  [_ A  /  x ]_ B
)
52, 4eqtrd 2398 . 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 1647    e. wcel 1715   _Vcvv 2873   [_csb 3167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-sbc 3078  df-csb 3168
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