Theorem csbov12g 5892

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

Assertion

Ref	Expression
csbov12g	|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )

Distinct variable group:   x, F

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

Proof of Theorem csbov12g

Step	Hyp	Ref	Expression
1		csbov123g 5891	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
2		csbconstg 3063	. . 3 |- ( A e. V -> [_ A / x ]_ F = F )
3	2	oveqd 5870	. 2 |- ( A e. V -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
4	1, 3	eqtrd 2203	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   [_csb 3049  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  csbov1g  5893  csbov2g  5894  fprodmul  11554  mulcncflem  13384