Theorem csbov1g 5928

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 5927	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 3083	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	oveq2d 5904	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( [_ A / x ]_ B F C ) )
4	1, 3	eqtrd 2220	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2158   [_csb 3069  (class class class)co 5888

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by:  modfsummodlemstep  11478  fprodmodd  11662