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Theorem csbnest1g 3133
Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
csbnest1g  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  x ]_ C )

Proof of Theorem csbnest1g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ C
21ax-gen 1533 . . 3  |-  A. y F/_ x [_ y  /  x ]_ C
3 csbnestgf 3129 . . 3  |-  ( ( A  e.  V  /\  A. y F/_ x [_ y  /  x ]_ C
)  ->  [_ A  /  x ]_ [_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
42, 3mpan2 652 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
5 csbco 3090 . . 3  |-  [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ B  /  x ]_ C
65csbeq2i 3107 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ A  /  x ]_ [_ B  /  x ]_ C
7 csbco 3090 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  x ]_ C
84, 6, 73eqtr3g 2338 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684   F/_wnfc 2406   [_csb 3081
This theorem is referenced by:  csbnest1gOLD  3134  csbidmg  3135
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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