Theorem csbfvg 5594

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 5593	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` [_ A / x ]_ x ) )
2		csbvarg 3108	. . 3 |- ( A e. C -> [_ A / x ]_ x = A )
3	2	fveq2d 5558	. 2 |- ( A e. C -> ( F ` [_ A / x ]_ x ) = ( F ` A ) )
4	1, 3	eqtrd 2226	1 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   [_csb 3080   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262

This theorem is referenced by:  ixpsnval  6755  ixpsnbasval  13962