Theorem csbfv2g 5597

Description: Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbfv2g	|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )

Distinct variable group:   x, F

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem csbfv2g

Step	Hyp	Ref	Expression
1		csbfv12g 5596	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
2		csbconstg 3098	. . 3 |- ( A e. C -> [_ A / x ]_ F = F )
3	2	fveq1d 5560	. 2 |- ( A e. C -> ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( F ` [_ A / x ]_ B ) )
4	1, 3	eqtrd 2229	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   [_csb 3084   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266

This theorem is referenced by:  csbfvg  5598  fsumabs  11630  fprodabs  11781  ixpsnbasval  14022