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Theorem csbnest1g 3239
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 3219 . . . 4  |-  F/_ x [_ y  /  x ]_ C
21ax-gen 1552 . . 3  |-  A. y F/_ x [_ y  /  x ]_ C
3 csbnestgf 3235 . . 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 653 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ [_ A  /  x ]_ B  /  y ]_ [_ y  /  x ]_ C )
5 csbco 3196 . . 3  |-  [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ B  /  x ]_ C
65csbeq2i 3213 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ A  /  x ]_ [_ B  /  x ]_ C
7 csbco 3196 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  x ]_ C
84, 6, 73eqtr3g 2435 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 1546    = wceq 1649    e. wcel 1717   F/_wnfc 2503   [_csb 3187
This theorem is referenced by:  csbidmg  3240
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-sbc 3098  df-csb 3188
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