MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbidmg Unicode version

Theorem csbidmg 3272
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 2932 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 csbnest1g 3271 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ [_ A  /  x ]_ A  /  x ]_ B )
3 csbconstg 3233 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ A  =  A )
43csbeq1d 3225 . . 3  |-  ( A  e.  _V  ->  [_ [_ A  /  x ]_ A  /  x ]_ B  =  [_ A  /  x ]_ B
)
52, 4eqtrd 2444 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
61, 5syl 16 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2924   [_csb 3219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-sbc 3130  df-csb 3220
  Copyright terms: Public domain W3C validator