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Theorem csbfv2g 5341
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 5340 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
2 csbconstg 2945 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ F  =  F )
32fveq1d 5307 . 2  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( F `  [_ A  /  x ]_ B ) )
41, 3eqtrd 2120 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( F `  [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   [_csb 2933   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023
This theorem is referenced by:  csbfvg  5342  fsumabs  10855
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