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

Step	Hyp	Ref	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 )
3	2	fveq2d 5322	. 2 |- ( A e. C -> ( F ` [_ A / x ]_ x ) = ( F ` A ) )
4	1, 3	eqtrd 2121	1 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )

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