Theorem sbcel12 3802

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 3302	. . . 4 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 3302	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2553	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 3302	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2553	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2501	. . . 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 2236	. . . . . . 7 |- F/ x [ z / x ] y e. B
8	7	nfab 2584	. . . . . 6 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 2236	. . . . . . 7 |- F/ x [ z / x ] y e. C
10	9	nfab 2584	. . . . . 6 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2593	. . . . 5 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2565	. . . . . 6 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2565	. . . . . 6 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2501	. . . . 5 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 2206	. . . 4 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 3140	. . 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 3396	. . . 4 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3396	. . . 4 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2500	. . 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 266	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
21		sbcex 3309	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
22	21	con3i 140	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
23		noel 3765	. . . 4 |- -. [_ A / x ]_ B e. (/)
24		csbprc 3800	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
25	24	eleq2d 2492	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. (/) ) )
26	23, 25	mtbiri 304	. . 3 |- ( -. A e. _V -> -. [_ A / x ]_ B e. [_ A / x ]_ C )
27	22, 26	2falsed 352	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
28	20, 27	pm2.61i 167	1 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 187   = wceq 1437   [wsb 1790   e. wcel 1872   {cab 2407   _Vcvv 3080   [.wsbc 3299   [_csb 3395   (/)c0 3761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762

This theorem is referenced by:  sbcnel12g  3804  sbcel1g  3806  sbcel2  3808  sbccsb2  3824  csbmpt12  4754  ixpsnval  7536  fmptdF  28257  csbmpt22g  31696  csbfinxpg  31744  finixpnum  31894