Theorem csbov1g 6069

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

Assertion

Ref	Expression
csbov1g	|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) )

Distinct variable groups:   x, C   x, F

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

Proof of Theorem csbov1g

Step	Hyp	Ref	Expression
1		csbov12g 6068	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 3142	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	oveq2d 6044	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( [_ A / x ]_ B F C ) )
4	1, 3	eqtrd 2264	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   [_csb 3128  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  modfsummodlemstep  12081  fprodmodd  12265