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Theorem csbafv12g 30184
Description: Move class substitution in and out of a function value, analogous to csbfv12gOLD 5828, with a direct proof proposed by Mario Carneiro, analogous to csbovgOLD 6226. (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 3392 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F''' B )  =  [_ A  /  x ]_ ( F''' B ) )
2 csbeq1 3392 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3392 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3afveq12d 30180 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F''' [_ y  /  x ]_ B )  =  (
[_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
51, 4eqeq12d 2473 . 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 3074 . . 3  |-  y  e. 
_V
7 nfcsb1v 3405 . . . 4  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3405 . . . 4  |-  F/_ x [_ y  /  x ]_ B
97, 8nfafv 30183 . . 3  |-  F/_ x
( [_ y  /  x ]_ F''' [_ y  /  x ]_ B )
10 csbeq1a 3398 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3398 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11afveq12d 30180 . . 3  |-  ( x  =  y  ->  ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B ) )
136, 9, 12csbief 3414 . 2  |-  [_ y  /  x ]_ ( F''' B )  =  (
[_ y  /  x ]_ F''' [_ y  /  x ]_ B )
145, 13vtoclg 3129 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 1370    e. wcel 1758   [_csb 3389  '''cafv 30159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-res 4953  df-iota 5482  df-fun 5521  df-fv 5527  df-dfat 30161  df-afv 30162
This theorem is referenced by: (None)
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