Theorem csbfv12g 5666

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 5310	. . 3 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [. A / x ]. B F y ) )
2		sbcbrg 4137	. . . . 5 |- ( A e. C -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y ) )
3		csbconstg 3138	. . . . . 6 |- ( A e. C -> [_ A / x ]_ y = y )
4	3	breq2d 4094	. . . . 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 5300	. . 3 |- ( A e. C -> ( iota y [. A / x ]. B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
7	1, 6	eqtrd 2262	. 2 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
8		df-fv 5325	. . 3 |- ( F ` B ) = ( iota y B F y )
9	8	csbeq2i 3151	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ ( iota y B F y )
10		df-fv 5325	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y )
11	7, 9, 10	3eqtr4g 2287	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   [.wsbc 3028   [_csb 3124   class class class wbr 4082   iotacio 5275   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325

This theorem is referenced by:  csbfv2g  5667