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Theorem sbcnestg 3302
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 1630 . . 3  |-  F/ x ph
21ax-gen 1556 . 2  |-  A. y F/ x ph
3 sbcnestgf 3300 . 2  |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
42, 3mpan2 654 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 178   A.wal 1550   F/wnf 1554    e. wcel 1726   [.wsbc 3163   [_csb 3253
This theorem is referenced by:  sbcco3g  3307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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