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Theorem csbfvg 5733
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 5732 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  [_ A  /  x ]_ x ) )
2 csbvarg 3270 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ x  =  A )
32fveq2d 5724 . 2  |-  ( A  e.  C  ->  ( F `  [_ A  /  x ]_ x )  =  ( F `  A
) )
41, 3eqtrd 2467 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   [_csb 3243   ` cfv 5446
This theorem is referenced by:  cdlemkid3N  31667  cdlemkid4  31668  cdlemk39s  31673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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