Theorem csbfvg 5681

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 5680	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` [_ A / x ]_ x ) )
2		csbvarg 3155	. . 3 |- ( A e. C -> [_ A / x ]_ x = A )
3	2	fveq2d 5643	. 2 |- ( A e. C -> ( F ` [_ A / x ]_ x ) = ( F ` A ) )
4	1, 3	eqtrd 2264	1 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   [_csb 3127   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334

This theorem is referenced by:  ixpsnval  6869  swrdspsleq  11247  ixpsnbasval  14479