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Theorem csbnest1g 3296
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 3276 . . . 4  |-  F/_ x [_ y  /  x ]_ C
21ax-gen 1555 . . 3  |-  A. y F/_ x [_ y  /  x ]_ C
3 csbnestgf 3292 . . 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 3253 . . 3  |-  [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ B  /  x ]_ C
65csbeq2i 3270 . 2  |-  [_ A  /  x ]_ [_ B  /  y ]_ [_ y  /  x ]_ C  = 
[_ A  /  x ]_ [_ B  /  x ]_ C
7 csbco 3253 . 2  |-  [_ [_ A  /  x ]_ B  / 
y ]_ [_ y  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  x ]_ C
84, 6, 73eqtr3g 2491 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 1549    = wceq 1652    e. wcel 1725   F/_wnfc 2559   [_csb 3244
This theorem is referenced by:  csbidmg  3297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951  df-sbc 3155  df-csb 3245
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