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Theorem csbafv12g 27979
Description: Move class substitution in and out of a function value, analogous to csbfv12g 5740, with a direct proof proposed by Mario Carneiro, analogous to csbovg 6114. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
csbafv12g  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
[_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )

Proof of Theorem csbafv12g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3256 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F''' B )  =  [_ A  /  x ]_ ( F''' B ) )
2 csbeq1 3256 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3256 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3afveq12d 27975 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F''' [_ y  /  x ]_ B )  =  (
[_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
51, 4eqeq12d 2452 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B )  <->  [_ A  /  x ]_ ( F''' B )  =  ( [_ A  /  x ]_ F''' [_ A  /  x ]_ B ) ) )
6 vex 2961 . . 3  |-  y  e. 
_V
7 nfcsb1v 3285 . . . 4  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3285 . . . 4  |-  F/_ x [_ y  /  x ]_ B
97, 8nfafv 27978 . . 3  |-  F/_ x
( [_ y  /  x ]_ F''' [_ y  /  x ]_ B )
10 csbeq1a 3261 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3261 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11afveq12d 27975 . . 3  |-  ( x  =  y  ->  ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B ) )
136, 9, 12csbief 3294 . 2  |-  [_ y  /  x ]_ ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B )
145, 13vtoclg 3013 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
[_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   [_csb 3253  '''cafv 27950
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-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-dfat 27952  df-afv 27953
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