Theorem csbov2g 5874

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Assertion

Ref	Expression
csbov2g	|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) )

Distinct variable groups:   x, B   x, F

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

Proof of Theorem csbov2g

Step	Hyp	Ref	Expression
1		csbov12g 5872	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 3054	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	oveq1d 5851	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( B F [_ A / x ]_ C ) )
4	1, 3	eqtrd 2197	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1342   e. wcel 2135   [_csb 3040  (class class class)co 5836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-iota 5147  df-fv 5190  df-ov 5839

This theorem is referenced by:  csbnegg  8087  fsummulc2  11375