Theorem sbcel12 3740

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 3238	. . . 4 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 3238	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2570	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 3238	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2570	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2524	. . . 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 2267	. . . . . . 7 |- F/ x [ z / x ] y e. B
8	7	nfab 2597	. . . . . 6 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 2267	. . . . . . 7 |- F/ x [ z / x ] y e. C
10	9	nfab 2597	. . . . . 6 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2605	. . . . 5 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2579	. . . . . 6 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2579	. . . . . 6 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2524	. . . . 5 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 2238	. . . 4 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 3076	. . 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 3332	. . . 4 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3332	. . . 4 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2523	. . 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 271	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
21		sbcex 3245	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
22	21	con3i 142	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
23		noel 3703	. . . 4 |- -. [_ A / x ]_ B e. (/)
24		csbprc 3738	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
25	24	eleq2d 2515	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. (/) ) )
26	23, 25	mtbiri 309	. . 3 |- ( -. A e. _V -> -. [_ A / x ]_ B e. [_ A / x ]_ C )
27	22, 26	2falsed 357	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
28	20, 27	pm2.61i 169	1 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 189   = wceq 1448   [wsb 1801   e. wcel 1891   {cab 2438   _Vcvv 3013   [.wsbc 3235   [_csb 3331   (/)c0 3699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-in 3379  df-ss 3386  df-nul 3700

This theorem is referenced by:  sbcnel12g  3742  sbcel1g  3744  sbcel2  3746  sbccsb2  3762  csbmpt12  4708  ixpsnval  7512  fmptdF  28264  csbmpt22g  31734  csbfinxpg  31782  finixpnum  31932