Theorem sbcel12 3786

Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcel12	|- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )

Proof of Theorem sbcel12

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3297	. . . 4 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 3297	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2590	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 3297	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2590	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2536	. . . 4 |- ( z = A -> ( { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } ) )
7		nfs1v 2151	. . . . . . 7 |- F/ x [ z / x ] y e. B
8	7	nfab 2620	. . . . . 6 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 2151	. . . . . . 7 |- F/ x [ z / x ] y e. C
10	9	nfab 2620	. . . . . 6 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2629	. . . . 5 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2601	. . . . . 6 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2601	. . . . . 6 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2536	. . . . 5 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 2110	. . . 4 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 3137	. . 3 |- ( A e. _V -> ( [. A / x ]. B e. C <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } ) )
17		df-csb 3399	. . . 4 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3399	. . . 4 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2533	. . 3 |- ( [_ A / x ]_ B e. [_ A / x ]_ C <-> { y | [. A / x ]. y e. B } e. { y | [. A / x ]. y e. C } )
20	16, 19	syl6bbr 263	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
21		sbcex 3304	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
22	21	con3i 135	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
23		noel 3752	. . . 4 |- -. [_ A / x ]_ B e. (/)
24		csbprc 3784	. . . . . 6 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
25	24	eleq2d 2524	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. (/) ) )
26	25	notbid 294	. . . 4 |- ( -. A e. _V -> ( -. [_ A / x ]_ B e. [_ A / x ]_ C <-> -. [_ A / x ]_ B e. (/) ) )
27	23, 26	mpbiri 233	. . 3 |- ( -. A e. _V -> -. [_ A / x ]_ B e. [_ A / x ]_ C )
28	22, 27	2falsed 351	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
29	20, 28	pm2.61i 164	1 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1370   [wsb 1702   e. wcel 1758   {cab 2439   _Vcvv 3078   [.wsbc 3294   [_csb 3398   (/)c0 3748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749

This theorem is referenced by:  sbcnel12g  3789  sbcel1g  3792  sbcel2  3794  sbccsb2  3814  csbmpt12  4733  ixpsnval  7379  fmptdF  26150  finixpnum  28585