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Theorem csbnestg 3217
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 2502 . . 3  |-  F/_ x C
21ax-gen 1551 . 2  |-  A. y F/_ x C
3 csbnestgf 3215 . 2  |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  / 
y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C
)
42, 3mpan2 652 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 1545    = wceq 1647    e. wcel 1715   F/_wnfc 2489   [_csb 3167
This theorem is referenced by:  csbnestgOLD  3218  csbco3g  3224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-sbc 3078  df-csb 3168
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