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Theorem csbnestg 3269
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 2548 . . 3  |-  F/_ x C
21ax-gen 1552 . 2  |-  A. y F/_ x C
3 csbnestgf 3267 . 2  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)
42, 3mpan2 653 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 1546    = wceq 1649    e. wcel 1721   F/_wnfc 2535   [_csb 3219
This theorem is referenced by:  csbnestgOLD  3270  csbco3g  3275  disjxpin  23989  cdleme31snd  30880
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-sbc 3130  df-csb 3220
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