Theorem csbov2g 6691

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 6689	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 3546	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	oveq1d 6665	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( B F [_ A / x ]_ C ) )
4	1, 3	eqtrd 2656	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  csbnegg  10278  prmgaplem7  15761  matgsum  20243  scmatscm  20319  pm2mpf1lem  20599  pm2mpcoe1  20605  pm2mpmhmlem2  20624  monmat2matmon  20629  divcncf  23216  logbmpt  24526  finxpreclem4  33231  cotrclrcl  38034  ply1mulgsumlem3  42176  ply1mulgsumlem4  42177  ply1mulgsum  42178