Theorem csbov2g 5744

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 5742	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 2967	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	oveq1d 5721	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( B F [_ A / x ]_ C ) )
4	1, 3	eqtrd 2132	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1299   e. wcel 1448   [_csb 2955  (class class class)co 5706

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709

This theorem is referenced by:  csbnegg  7831  fsummulc2  11056