Theorem csbfv12g 5465

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 5124	. . 3 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [. A / x ]. B F y ) )
2		sbcbrg 3990	. . . . 5 |- ( A e. C -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y ) )
3		csbconstg 3021	. . . . . 6 |- ( A e. C -> [_ A / x ]_ y = y )
4	3	breq2d 3949	. . . . 5 |- ( A e. C -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y <-> [_ A / x ]_ B [_ A / x ]_ F y ) )
5	2, 4	bitrd 187	. . . 4 |- ( A e. C -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F y ) )
6	5	iotabidv 5117	. . 3 |- ( A e. C -> ( iota y [. A / x ]. B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
7	1, 6	eqtrd 2173	. 2 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
8		df-fv 5139	. . 3 |- ( F ` B ) = ( iota y B F y )
9	8	csbeq2i 3034	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ ( iota y B F y )
10		df-fv 5139	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y )
11	7, 9, 10	3eqtr4g 2198	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   [.wsbc 2913   [_csb 3007   class class class wbr 3937   iotacio 5094   ` cfv 5131

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139

This theorem is referenced by:  csbfv2g  5466