Theorem csbriota 6059

Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)

Assertion

Ref	Expression
csbriota	|- [_ 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)

Proof of Theorem csbriota

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3286	. . . 4 |- ( z = A -> [_ z / x ]_ ( iota_ y e. B ph ) = [_ A / x ]_ ( iota_ y e. B ph ) )
2		dfsbcq2 3184	. . . . 5 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	riotabidv 6049	. . . 4 |- ( z = A -> ( iota_ y e. B [ z / x ] ph ) = ( iota_ y e. B [. A / x ]. ph ) )
4	1, 3	eqeq12d 2452	. . 3 |- ( 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 2970	. . . 4 |- z e. _V
6		nfs1v 2142	. . . . 5 |- F/ x [ z / x ] ph
7		nfcv 2574	. . . . 5 |- F/_ x B
8	6, 7	nfriota 6056	. . . 4 |- F/_ x ( iota_ y e. B [ z / x ] ph )
9		sbequ12 1936	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
10	9	riotabidv 6049	. . . 4 |- ( x = z -> ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) )
11	5, 8, 10	csbief 3308	. . 3 |- [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph )
12	4, 11	vtoclg 3025	. 2 |- ( A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
13		csbprc 3668	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = (/) )
14		df-riota 6047	. . . 4 |- ( iota_ y e. B [. A / x ]. ph ) = ( iota y ( y e. B /\ [. A / x ]. ph ) )
15		euex 2279	. . . . . . 7 |- ( E! y ( y e. B /\ [. A / x ]. ph ) -> E. y ( y e. B /\ [. A / x ]. ph ) )
16		sbcex 3191	. . . . . . . . 9 |- ( [. A / x ]. ph -> A e. _V )
17	16	adantl 466	. . . . . . . 8 |- ( ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
18	17	exlimiv 1688	. . . . . . 7 |- ( E. y ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
19	15, 18	syl 16	. . . . . 6 |- ( E! y ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
20	19	con3i 135	. . . . 5 |- ( -. A e. _V -> -. E! y ( y e. B /\ [. A / x ]. ph ) )
21		iotanul 5391	. . . . 5 |- ( -. E! y ( y e. B /\ [. A / x ]. ph ) -> ( iota y ( y e. B /\ [. A / x ]. ph ) ) = (/) )
22	20, 21	syl 16	. . . 4 |- ( -. A e. _V -> ( iota y ( y e. B /\ [. A / x ]. ph ) ) = (/) )
23	14, 22	syl5req 2483	. . 3 |- ( -. A e. _V -> (/) = ( iota_ y e. B [. A / x ]. ph ) )
24	13, 23	eqtrd 2470	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
25	12, 24	pm2.61i 164	1 |- [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph )

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1369   E.wex 1586   [wsb 1700   e. wcel 1756   E!weu 2252   _Vcvv 2967   [.wsbc 3181   [_csb 3283   (/)c0 3632   iotacio 5374   iota_crio 6046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-in 3330  df-ss 3337  df-nul 3633  df-sn 3873  df-uni 4087  df-iota 5376  df-riota 6047

This theorem is referenced by:  csbriotagOLD  6060  cdlemkid3N  34417  cdlemkid4  34418