Theorem csbov1g 5545

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 5544	. 2 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
2		csbconstg 2864	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	oveq2d 5528	. 2 |- ( A e. V -> ( [_ A / x ]_ B F [_ A / x ]_ C ) = ( [_ A / x ]_ B F C ) )
4	1, 3	eqtrd 2072	1 |- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   [_csb 2852  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by: (None)