Theorem sbcel2 3839

Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcel2	|- ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem sbcel2

Step	Hyp	Ref	Expression
1		sbcel12 3832	. . 3 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3443	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3	2	eleq1d 2526	. . 3 |- ( A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) )
4	1, 3	syl5bb 257	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
5		sbcex 3337	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
6	5	con3i 135	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
7		noel 3797	. . . 4 |- -. B e. (/)
8		csbprc 3830	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
9	8	eleq2d 2527	. . . 4 |- ( -. A e. _V -> ( B e. [_ A / x ]_ C <-> B e. (/) ) )
10	7, 9	mtbiri 303	. . 3 |- ( -. A e. _V -> -. B e. [_ A / x ]_ C )
11	6, 10	2falsed 351	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
12	4, 11	pm2.61i 164	1 |- ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 184   e. wcel 1819   _Vcvv 3109   [.wsbc 3327   [_csb 3430   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794

This theorem is referenced by:  csbcom  3845  sbccsb  3856  sbnfc2  3859  csbab  3860  sbcssg  3943  csbuni  4279  csbxp  5090  csbdm  5207  issubc  15251  nbgraopALT  24551  esum2dlem  28264  csbcom2fi  30718  bj-sbeq  34590  bj-sbceqgALT  34591  bj-sels  34642