Theorem csbfv12g 5551

Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbfv12g	|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Proof of Theorem csbfv12g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbiotag 5209	. . 3 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [. A / x ]. B F y ) )
2		sbcbrg 4057	. . . . 5 |- ( A e. C -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y ) )
3		csbconstg 3071	. . . . . 6 |- ( A e. C -> [_ A / x ]_ y = y )
4	3	breq2d 4015	. . . . 5 |- ( A e. C -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y <-> [_ A / x ]_ B [_ A / x ]_ F y ) )
5	2, 4	bitrd 188	. . . 4 |- ( A e. C -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F y ) )
6	5	iotabidv 5199	. . 3 |- ( A e. C -> ( iota y [. A / x ]. B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
7	1, 6	eqtrd 2210	. 2 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
8		df-fv 5224	. . 3 |- ( F ` B ) = ( iota y B F y )
9	8	csbeq2i 3084	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ ( iota y B F y )
10		df-fv 5224	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y )
11	7, 9, 10	3eqtr4g 2235	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   [.wsbc 2962   [_csb 3057   class class class wbr 4003   iotacio 5176   ` cfv 5216

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 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224

This theorem is referenced by:  csbfv2g  5552