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Theorem csbfv2g 5828
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 5826 . 2  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )
2 csbconstg 3401 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ F  =  F )
32fveq1d 5793 . 2  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( F `  [_ A  /  x ]_ B ) )
41, 3syl5eq 2504 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( F `  [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   [_csb 3388   ` cfv 5518
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4521  ax-pow 4570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-dm 4950  df-iota 5481  df-fv 5526
This theorem is referenced by:  csbfv  5829  csbfvgOLD  5830  ixpsnval  7368  swrdspsleq  12446  sumeq2ii  13274  fsumabs  13368  ixpsnbasval  17398  evl1gsumdlem  17901  iuninc  26047  prodeq2ii  27562  fprodabs  27620  coe1fzgsumdlem  30981  pmat2matp  31281  cayhamlem4  31345  cdlemk39s  34891
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