Theorem csbov1g 5918

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 5917	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 3073	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	oveq2d 5894	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( [_ A / x ]_ B F C ) )
4	1, 3	eqtrd 2210	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   [_csb 3059  (class class class)co 5878

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5881

This theorem is referenced by:  modfsummodlemstep  11468  fprodmodd  11652