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Theorem sbcnestg 3143
Description: Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestg  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ph( y)    A( x, y)    B( x, y)    V( x, y)

Proof of Theorem sbcnestg
StepHypRef Expression
1 nfv 1609 . . 3  |-  F/ x ph
21ax-gen 1536 . 2  |-  A. y F/ x ph
3 sbcnestgf 3141 . 2  |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
42, 3mpan2 652 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   F/wnf 1534    e. wcel 1696   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  sbcco3g  3149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
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