Theorem csbriotag 5967

Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)

Assertion

Ref	Expression
csbriotag	|- ( A e. V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )

Distinct variable groups:   y, A   x, B   x, y

Allowed substitution hints:   ph( x, y)   A( x)   B( y)   V( x, y)

Proof of Theorem csbriotag

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3127	. . 3 |- ( z = A -> [_ z / x ]_ ( iota_ y e. B ph ) = [_ A / x ]_ ( iota_ y e. B ph ) )
2		dfsbcq2 3031	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	riotabidv 5955	. . 3 |- ( z = A -> ( iota_ y e. B [ z / x ] ph ) = ( iota_ y e. B [. A / x ]. ph ) )
4	1, 3	eqeq12d 2244	. 2 |- ( z = A -> ( [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) <-> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) ) )
5		vex 2802	. . 3 |- z e. _V
6		nfs1v 1990	. . . 4 |- F/ x [ z / x ] ph
7		nfcv 2372	. . . 4 |- F/_ x B
8	6, 7	nfriota 5963	. . 3 |- F/_ x ( iota_ y e. B [ z / x ] ph )
9		sbequ12 1817	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
10	9	riotabidv 5955	. . 3 |- ( x = z -> ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) )
11	5, 8, 10	csbief 3169	. 2 |- [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph )
12	4, 11	vtoclg 2861	1 |- ( A e. V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1395   [wsb 1808   e. wcel 2200   [.wsbc 3028   [_csb 3124   iota_crio 5952

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-sn 3672  df-uni 3888  df-iota 5277  df-riota 5953

This theorem is referenced by: (None)