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Theorem csbafv12g 32461
Description: Move class substitution in and out of a function value, analogous to csbfv12 5883, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6304. (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 3423 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F''' B )  =  [_ A  /  x ]_ ( F''' B ) )
2 csbeq1 3423 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3423 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3afveq12d 32457 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F''' [_ y  /  x ]_ B )  =  (
[_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
51, 4eqeq12d 2476 . 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 3109 . . 3  |-  y  e. 
_V
7 nfcsb1v 3436 . . . 4  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3436 . . . 4  |-  F/_ x [_ y  /  x ]_ B
97, 8nfafv 32460 . . 3  |-  F/_ x
( [_ y  /  x ]_ F''' [_ y  /  x ]_ B )
10 csbeq1a 3429 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3429 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11afveq12d 32457 . . 3  |-  ( x  =  y  ->  ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B ) )
136, 9, 12csbief 3445 . 2  |-  [_ y  /  x ]_ ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B )
145, 13vtoclg 3164 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
[_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   [_csb 3420  '''cafv 32438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-dfat 32440  df-afv 32441
This theorem is referenced by: (None)
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