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Theorem csbfv2g 5909
Description: Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbfv2g  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( F `  [_ A  /  x ]_ B ) )
Distinct variable group:    x, F
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem csbfv2g
StepHypRef Expression
1 csbfv12 5907 . 2  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )
2 csbconstg 3453 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ F  =  F )
32fveq1d 5874 . 2  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( F `  [_ A  /  x ]_ B ) )
41, 3syl5eq 2520 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( F `  [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   [_csb 3440   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582  ax-pow 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602
This theorem is referenced by:  csbfv  5910  csbfvgOLD  5911  ixpsnval  7484  swrdspsleq  12653  sumeq2ii  13495  fsumabs  13595  ixpsnbasval  17726  coe1fzgsumdlem  18213  evl1gsumdlem  18262  pm2mp  19195  cayhamlem4  19258  nbgraopALT  24247  iuninc  27251  prodeq2ii  28972  fprodabs  29030  cdlemk39s  36136
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