Theorem csbfvg 5639

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 5638	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` [_ A / x ]_ x ) )
2		csbvarg 3129	. . 3 |- ( A e. C -> [_ A / x ]_ x = A )
3	2	fveq2d 5603	. 2 |- ( A e. C -> ( F ` [_ A / x ]_ x ) = ( F ` A ) )
4	1, 3	eqtrd 2240	1 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   [_csb 3101   ` cfv 5290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298

This theorem is referenced by:  ixpsnval  6811  swrdspsleq  11158  ixpsnbasval  14343