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Theorem csbfvg 5427
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 5426 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  [_ A  /  x ]_ x ) )
2 csbvarg 3000 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ x  =  A )
32fveq2d 5393 . 2  |-  ( A  e.  C  ->  ( F `  [_ A  /  x ]_ x )  =  ( F `  A
) )
41, 3eqtrd 2150 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   [_csb 2975   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101
This theorem is referenced by:  ixpsnval  6563
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