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Theorem csbfvg 5538
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 5537 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  [_ A  /  x ]_ x ) )
2 csbvarg 3108 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ x  =  A )
32fveq2d 5529 . 2  |-  ( A  e.  C  ->  ( F `  [_ A  /  x ]_ x )  =  ( F `  A
) )
41, 3eqtrd 2315 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   [_csb 3081   ` cfv 5255
This theorem is referenced by:  cdlemkid3N  31122  cdlemkid4  31123  cdlemk39s  31128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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