Theorem sbcel12 3983

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 3438	. . . 4 |- ( z = A -> ( [ z / x ] B e. C <-> [. A / x ]. B e. C ) )
2		dfsbcq2 3438	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. B <-> [. A / x ]. y e. B ) )
3	2	abbidv 2741	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. B } = { y | [. A / x ]. y e. B } )
4		dfsbcq2 3438	. . . . . 6 |- ( z = A -> ( [ z / x ] y e. C <-> [. A / x ]. y e. C ) )
5	4	abbidv 2741	. . . . 5 |- ( z = A -> { y | [ z / x ] y e. C } = { y | [. A / x ]. y e. C } )
6	3, 5	eleq12d 2695	. . . 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 2437	. . . . . . 7 |- F/ x [ z / x ] y e. B
8	7	nfab 2769	. . . . . 6 |- F/_ x { y | [ z / x ] y e. B }
9		nfs1v 2437	. . . . . . 7 |- F/ x [ z / x ] y e. C
10	9	nfab 2769	. . . . . 6 |- F/_ x { y | [ z / x ] y e. C }
11	8, 10	nfel 2777	. . . . 5 |- F/ x { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C }
12		sbab 2750	. . . . . 6 |- ( x = z -> B = { y | [ z / x ] y e. B } )
13		sbab 2750	. . . . . 6 |- ( x = z -> C = { y | [ z / x ] y e. C } )
14	12, 13	eleq12d 2695	. . . . 5 |- ( x = z -> ( B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } ) )
15	11, 14	sbie 2408	. . . 4 |- ( [ z / x ] B e. C <-> { y | [ z / x ] y e. B } e. { y | [ z / x ] y e. C } )
16	1, 6, 15	vtoclbg 3267	. . 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 3534	. . . 4 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
18		df-csb 3534	. . . 4 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
19	17, 18	eleq12i 2694	. . 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 278	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
21		sbcex 3445	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
22	21	con3i 150	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
23		noel 3919	. . . 4 |- -. [_ A / x ]_ B e. (/)
24		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
25	24	eleq2d 2687	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. (/) ) )
26	23, 25	mtbiri 317	. . 3 |- ( -. A e. _V -> -. [_ A / x ]_ B e. [_ A / x ]_ C )
27	22, 26	2falsed 366	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
28	20, 27	pm2.61i 176	1 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915

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-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  sbcnel12g  3985  sbcel1g  3987  sbcel2  3989  sbccsb2  4005  csbmpt12  5010  ixpsnval  7911  fmptdF  29456  csbmpt22g  33177  csbfinxpg  33225  finixpnum  33394