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Theorem sbcnestgf 3241
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf  |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )

Proof of Theorem sbcnestgf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3106 . . . . 5  |-  ( z  =  A  ->  ( [. z  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
2 csbeq1 3197 . . . . . 6  |-  ( z  =  A  ->  [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 dfsbcq 3106 . . . . . 6  |-  ( [_ z  /  x ]_ B  =  [_ A  /  x ]_ B  ->  ( [. [_ z  /  x ]_ B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
42, 3syl 16 . . . . 5  |-  ( z  =  A  ->  ( [. [_ z  /  x ]_ B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
51, 4bibi12d 313 . . . 4  |-  ( z  =  A  ->  (
( [. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph )  <->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) ) )
65imbi2d 308 . . 3  |-  ( z  =  A  ->  (
( A. y F/ x ph  ->  ( [. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )  <->  ( A. y F/ x ph  ->  (
[. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) ) ) )
7 vex 2902 . . . . 5  |-  z  e. 
_V
87a1i 11 . . . 4  |-  ( A. y F/ x ph  ->  z  e.  _V )
9 csbeq1a 3202 . . . . . 6  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
10 dfsbcq 3106 . . . . . 6  |-  ( B  =  [_ z  /  x ]_ B  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
119, 10syl 16 . . . . 5  |-  ( x  =  z  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
1211adantl 453 . . . 4  |-  ( ( A. y F/ x ph  /\  x  =  z )  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  / 
y ]. ph ) )
13 nfnf1 1798 . . . . 5  |-  F/ x F/ x ph
1413nfal 1854 . . . 4  |-  F/ x A. y F/ x ph
15 nfa1 1796 . . . . 5  |-  F/ y A. y F/ x ph
16 nfcsb1v 3226 . . . . . 6  |-  F/_ x [_ z  /  x ]_ B
1716a1i 11 . . . . 5  |-  ( A. y F/ x ph  ->  F/_ x [_ z  /  x ]_ B )
18 sp 1755 . . . . 5  |-  ( A. y F/ x ph  ->  F/ x ph )
1915, 17, 18nfsbcd 3124 . . . 4  |-  ( A. y F/ x ph  ->  F/ x [. [_ z  /  x ]_ B  / 
y ]. ph )
208, 12, 14, 19sbciedf 3139 . . 3  |-  ( A. y F/ x ph  ->  (
[. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
216, 20vtoclg 2954 . 2  |-  ( A  e.  V  ->  ( A. y F/ x ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) ) )
2221imp 419 1  |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2510   _Vcvv 2899   [.wsbc 3104   [_csb 3194
This theorem is referenced by:  csbnestgf  3242  sbcnestg  3243
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105  df-csb 3195
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