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Theorem csbfvg 5674
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 5673 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  [_ A  /  x ]_ x ) )
2 csbvarg 3214 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ x  =  A )
32fveq2d 5665 . 2  |-  ( A  e.  C  ->  ( F `  [_ A  /  x ]_ x )  =  ( F `  A
) )
41, 3eqtrd 2412 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   [_csb 3187   ` cfv 5387
This theorem is referenced by:  cdlemkid3N  31098  cdlemkid4  31099  cdlemk39s  31104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395
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