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Theorem sbcnestgf 3290
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 3155 . . . . 5  |-  ( z  =  A  ->  ( [. z  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
2 csbeq1 3246 . . . . . 6  |-  ( z  =  A  ->  [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 dfsbcq 3155 . . . . . 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 2951 . . . . 5  |-  z  e. 
_V
87a1i 11 . . . 4  |-  ( A. y F/ x ph  ->  z  e.  _V )
9 csbeq1a 3251 . . . . . 6  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
10 dfsbcq 3155 . . . . . 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 1808 . . . . 5  |-  F/ x F/ x ph
1413nfal 1864 . . . 4  |-  F/ x A. y F/ x ph
15 nfa1 1806 . . . . 5  |-  F/ y A. y F/ x ph
16 nfcsb1v 3275 . . . . . 6  |-  F/_ x [_ z  /  x ]_ B
1716a1i 11 . . . . 5  |-  ( A. y F/ x ph  ->  F/_ x [_ z  /  x ]_ B )
18 sp 1763 . . . . 5  |-  ( A. y F/ x ph  ->  F/ x ph )
1915, 17, 18nfsbcd 3173 . . . 4  |-  ( A. y F/ x ph  ->  F/ x [. [_ z  /  x ]_ B  / 
y ]. ph )
208, 12, 14, 19sbciedf 3188 . . 3  |-  ( A. y F/ x ph  ->  (
[. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
216, 20vtoclg 3003 . 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 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558   _Vcvv 2948   [.wsbc 3153   [_csb 3243
This theorem is referenced by:  csbnestgf  3291  sbcnestg  3292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244
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