Theorem csbfv2g 5642

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 5641	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
2		csbconstg 3118	. . 3 |- ( A e. C -> [_ A / x ]_ F = F )
3	2	fveq1d 5605	. 2 |- ( A e. C -> ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( F ` [_ A / x ]_ B ) )
4	1, 3	eqtrd 2242	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1375   e. wcel 2180   [_csb 3104   ` cfv 5294

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-iota 5254  df-fv 5302

This theorem is referenced by:  csbfvg  5643  fsumabs  11942  fprodabs  12093  ixpsnbasval  14395