Theorem csbriota 6623

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 3536	. . . 4 |- ( z = A -> [_ z / x ]_ ( iota_ y e. B ph ) = [_ A / x ]_ ( iota_ y e. B ph ) )
2		dfsbcq2 3438	. . . . 5 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	riotabidv 6613	. . . 4 |- ( z = A -> ( iota_ y e. B [ z / x ] ph ) = ( iota_ y e. B [. A / x ]. ph ) )
4	1, 3	eqeq12d 2637	. . 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 3203	. . . 4 |- z e. _V
6		nfs1v 2437	. . . . 5 |- F/ x [ z / x ] ph
7		nfcv 2764	. . . . 5 |- F/_ x B
8	6, 7	nfriota 6620	. . . 4 |- F/_ x ( iota_ y e. B [ z / x ] ph )
9		sbequ12 2111	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
10	9	riotabidv 6613	. . . 4 |- ( x = z -> ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) )
11	5, 8, 10	csbief 3558	. . 3 |- [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph )
12	4, 11	vtoclg 3266	. 2 |- ( A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
13		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = (/) )
14		df-riota 6611	. . . 4 |- ( iota_ y e. B [. A / x ]. ph ) = ( iota y ( y e. B /\ [. A / x ]. ph ) )
15		euex 2494	. . . . . . 7 |- ( E! y ( y e. B /\ [. A / x ]. ph ) -> E. y ( y e. B /\ [. A / x ]. ph ) )
16		sbcex 3445	. . . . . . . . 9 |- ( [. A / x ]. ph -> A e. _V )
17	16	adantl 482	. . . . . . . 8 |- ( ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
18	17	exlimiv 1858	. . . . . . 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 150	. . . . 5 |- ( -. A e. _V -> -. E! y ( y e. B /\ [. A / x ]. ph ) )
21		iotanul 5866	. . . . 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 2669	. . 3 |- ( -. A e. _V -> (/) = ( iota_ y e. B [. A / x ]. ph ) )
24	13, 23	eqtrd 2656	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
25	12, 24	pm2.61i 176	1 |- [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   E!weu 2470   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   iotacio 5849   iota_crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by:  cdlemkid3N  36221  cdlemkid4  36222