Theorem csbriota 6289

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 3378	. . . 4 |- ( z = A -> [_ z / x ]_ ( iota_ y e. B ph ) = [_ A / x ]_ ( iota_ y e. B ph ) )
2		dfsbcq2 3282	. . . . 5 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	riotabidv 6279	. . . 4 |- ( z = A -> ( iota_ y e. B [ z / x ] ph ) = ( iota_ y e. B [. A / x ]. ph ) )
4	1, 3	eqeq12d 2477	. . 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 3060	. . . 4 |- z e. _V
6		nfs1v 2277	. . . . 5 |- F/ x [ z / x ] ph
7		nfcv 2603	. . . . 5 |- F/_ x B
8	6, 7	nfriota 6286	. . . 4 |- F/_ x ( iota_ y e. B [ z / x ] ph )
9		sbequ12 2094	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
10	9	riotabidv 6279	. . . 4 |- ( x = z -> ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) )
11	5, 8, 10	csbief 3400	. . 3 |- [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph )
12	4, 11	vtoclg 3119	. 2 |- ( A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
13		csbprc 3782	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = (/) )
14		df-riota 6277	. . . 4 |- ( iota_ y e. B [. A / x ]. ph ) = ( iota y ( y e. B /\ [. A / x ]. ph ) )
15		euex 2334	. . . . . . 7 |- ( E! y ( y e. B /\ [. A / x ]. ph ) -> E. y ( y e. B /\ [. A / x ]. ph ) )
16		sbcex 3289	. . . . . . . . 9 |- ( [. A / x ]. ph -> A e. _V )
17	16	adantl 472	. . . . . . . 8 |- ( ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
18	17	exlimiv 1787	. . . . . . 7 |- ( E. y ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
19	15, 18	syl 17	. . . . . 6 |- ( E! y ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
20	19	con3i 142	. . . . 5 |- ( -. A e. _V -> -. E! y ( y e. B /\ [. A / x ]. ph ) )
21		iotanul 5580	. . . . 5 |- ( -. E! y ( y e. B /\ [. A / x ]. ph ) -> ( iota y ( y e. B /\ [. A / x ]. ph ) ) = (/) )
22	20, 21	syl 17	. . . 4 |- ( -. A e. _V -> ( iota y ( y e. B /\ [. A / x ]. ph ) ) = (/) )
23	14, 22	syl5req 2509	. . 3 |- ( -. A e. _V -> (/) = ( iota_ y e. B [. A / x ]. ph ) )
24	13, 23	eqtrd 2496	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
25	12, 24	pm2.61i 169	1 |- [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph )

Syntax hints:   -. wn 3   /\ wa 375   = wceq 1455   E.wex 1674   [wsb 1808   e. wcel 1898   E!weu 2310   _Vcvv 3057   [.wsbc 3279   [_csb 3375   (/)c0 3743   iotacio 5563   iota_crio 6276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744  df-sn 3981  df-uni 4213  df-iota 5565  df-riota 6277

This theorem is referenced by:  cdlemkid3N  34545  cdlemkid4  34546