Theorem csbfv12g 5627

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 5273	. . 3 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [. A / x ]. B F y ) )
2		sbcbrg 4106	. . . . 5 |- ( A e. C -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y ) )
3		csbconstg 3111	. . . . . 6 |- ( A e. C -> [_ A / x ]_ y = y )
4	3	breq2d 4063	. . . . 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 5263	. . 3 |- ( A e. C -> ( iota y [. A / x ]. B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
7	1, 6	eqtrd 2239	. 2 |- ( A e. C -> [_ A / x ]_ ( iota y B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
8		df-fv 5288	. . 3 |- ( F ` B ) = ( iota y B F y )
9	8	csbeq2i 3124	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ ( iota y B F y )
10		df-fv 5288	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y )
11	7, 9, 10	3eqtr4g 2264	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   [.wsbc 3002   [_csb 3097   class class class wbr 4051   iotacio 5239   ` cfv 5280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288

This theorem is referenced by:  csbfv2g  5628