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Theorem csbnestg 3111
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 2319 . . 3  |-  F/_ x C
21ax-gen 1449 . 2  |-  A. y F/_ x C
3 csbnestgf 3109 . 2  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)
42, 3mpan2 425 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 1351    = wceq 1353    e. wcel 2148   F/_wnfc 2306   [_csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by:  csbco3g  3115
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