Theorem csbov12g 5881

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

Assertion

Ref	Expression
csbov12g	|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )

Distinct variable group:   x, F

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

Proof of Theorem csbov12g

Step	Hyp	Ref	Expression
1		csbov123g 5880	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
2		csbconstg 3059	. . 3 |- ( A e. V -> [_ A / x ]_ F = F )
3	2	oveqd 5859	. 2 |- ( A e. V -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
4	1, 3	eqtrd 2198	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   [_csb 3045  (class class class)co 5842

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  csbov1g  5882  csbov2g  5883  fprodmul  11532  mulcncflem  13230