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Theorem csbnestg 3669
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 2578 . . 3  |-  F/_ x C
21ax-gen 1556 . 2  |-  A. y F/_ x C
3 csbnestgf 3667 . 2  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)
42, 3mpan2 654 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 1550    = wceq 1653    e. wcel 1727   F/_wnfc 2565   [_csb 3267
This theorem is referenced by:  csbnestgOLD  3670  csbco3g  3673  disjxpin  24059  cdleme31snd  31281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964  df-sbc 3168  df-csb 3268
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