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Theorem csbfv2g 5810
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 5809 . 2  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )
2 csbconstg 3361 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ F  =  F )
32fveq1d 5776 . 2  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( F `  [_ A  /  x ]_ B ) )
41, 3syl5eq 2435 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( F `  [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   [_csb 3348   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-nul 4496  ax-pow 4543
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-dm 4923  df-iota 5460  df-fv 5504
This theorem is referenced by:  csbfv  5811  ixpsnval  7391  swrdspsleq  12585  sumeq2ii  13517  fsumabs  13617  prodeq2ii  13722  fprodabs  13780  ixpsnbasval  17968  coe1fzgsumdlem  18456  evl1gsumdlem  18505  pm2mp  19411  cayhamlem4  19474  nbgraopALT  24545  iuninc  27557  cdlemk39s  37078
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