Theorem sbcel12 3828

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 3339	. . . 4 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 3339	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2603	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 3339	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2603	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2549	. . . 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 2164	. . . . . . 7 |- F/ x [ z / x ] y e. B
8	7	nfab 2633	. . . . . 6 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 2164	. . . . . . 7 |- F/ x [ z / x ] y e. C
10	9	nfab 2633	. . . . . 6 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2642	. . . . 5 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2614	. . . . . 6 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2614	. . . . . 6 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2549	. . . . 5 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 2123	. . . 4 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 3177	. . 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 3441	. . . 4 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3441	. . . 4 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2546	. . 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 3346	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
22	21	con3i 135	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
23		noel 3794	. . . 4 |- -. [_ A / x ]_ B e. (/)
24		csbprc 3826	. . . . . 6 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
25	24	eleq2d 2537	. . . . 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 1379   [wsb 1711   e. wcel 1767   {cab 2452   _Vcvv 3118   [.wsbc 3336   [_csb 3440   (/)c0 3790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791

This theorem is referenced by:  sbcnel12g  3831  sbcel1g  3834  sbcel2  3836  sbccsb2  3856  csbmpt12  4787  ixpsnval  7484  fmptdF  27318  finixpnum  29965