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Theorem sbcnestgf 2954
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 2818 . . . . 5  |-  ( z  =  A  ->  ( [. z  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
2 csbeq1 2912 . . . . . 6  |-  ( z  =  A  ->  [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 dfsbcq 2818 . . . . . 6  |-  ( [_ z  /  x ]_ B  =  [_ A  /  x ]_ B  ->  ( [. [_ z  /  x ]_ B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
42, 3syl 14 . . . . 5  |-  ( z  =  A  ->  ( [. [_ z  /  x ]_ B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
51, 4bibi12d 233 . . . 4  |-  ( z  =  A  ->  (
( [. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph )  <->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) ) )
65imbi2d 228 . . 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 2605 . . . . 5  |-  z  e. 
_V
87a1i 9 . . . 4  |-  ( A. y F/ x ph  ->  z  e.  _V )
9 csbeq1a 2917 . . . . . 6  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
10 dfsbcq 2818 . . . . . 6  |-  ( B  =  [_ z  /  x ]_ B  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
119, 10syl 14 . . . . 5  |-  ( x  =  z  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
1211adantl 271 . . . 4  |-  ( ( A. y F/ x ph  /\  x  =  z )  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  / 
y ]. ph ) )
13 nfnf1 1477 . . . . 5  |-  F/ x F/ x ph
1413nfal 1509 . . . 4  |-  F/ x A. y F/ x ph
15 nfa1 1475 . . . . 5  |-  F/ y A. y F/ x ph
16 nfcsb1v 2939 . . . . . 6  |-  F/_ x [_ z  /  x ]_ B
1716a1i 9 . . . . 5  |-  ( A. y F/ x ph  ->  F/_ x [_ z  /  x ]_ B )
18 sp 1442 . . . . 5  |-  ( A. y F/ x ph  ->  F/ x ph )
1915, 17, 18nfsbcd 2835 . . . 4  |-  ( A. y F/ x ph  ->  F/ x [. [_ z  /  x ]_ B  / 
y ]. ph )
208, 12, 14, 19sbciedf 2850 . . 3  |-  ( A. y F/ x ph  ->  (
[. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
216, 20vtoclg 2659 . 2  |-  ( A  e.  V  ->  ( A. y F/ x ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) ) )
2221imp 122 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    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   F/wnf 1390    e. wcel 1434   F/_wnfc 2207   _Vcvv 2602   [.wsbc 2816   [_csb 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910
This theorem is referenced by:  csbnestgf  2955  sbcnestg  2956
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