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Theorem csbfvg 5355
Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
Assertion
Ref Expression
csbfvg  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
Distinct variable group:    x, F
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem csbfvg
StepHypRef Expression
1 csbfv2g 5354 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  [_ A  /  x ]_ x ) )
2 csbvarg 2959 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ x  =  A )
32fveq2d 5322 . 2  |-  ( A  e.  C  ->  ( F `  [_ A  /  x ]_ x )  =  ( F `  A
) )
41, 3eqtrd 2121 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439   [_csb 2934   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036
This theorem is referenced by:  ixpsnval  6472
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