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Theorem csbnestg 2928
Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
Assertion
Ref Expression
csbnestg  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C )
Distinct variable group:    x, C
Allowed substitution hints:    A( x, y)    B( x, y)    C( y)    V( x, y)

Proof of Theorem csbnestg
StepHypRef Expression
1 nfcv 2194 . . 3  |-  F/_ x C
21ax-gen 1354 . 2  |-  A. y F/_ x C
3 csbnestgf 2926 . 2  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)
42, 3mpan2 409 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259    e. wcel 1409   F/_wnfc 2181   [_csb 2880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881
This theorem is referenced by:  csbco3g  2932
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