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Theorem csbfv2g 5707
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 csbfv12g 5705 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
2 csbconstg 3233 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ F  =  F )
32fveq1d 5697 . 2  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( F `  [_ A  /  x ]_ B ) )
41, 3eqtrd 2444 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( F `  [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   [_csb 3219   ` cfv 5421
This theorem is referenced by:  csbfvg  5708  sumeq2ii  12450  fsumabs  12543  iuninc  23972  prodeq2ii  25200  fprodabs  25258  cdlemk39s  31433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429
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