Theorem sbcel2 3836

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 3828	. . 3 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3453	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3	2	eleq1d 2536	. . 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 3346	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
6	5	con3i 135	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
7		noel 3794	. . . 4 |- -. B e. (/)
8		csbprc 3826	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
9	8	eleq2d 2537	. . . 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 1767   _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:  csbcom  3842  sbccsb  3854  sbnfc2  3859  csbab  3860  sbcssg  3944  csbuni  4279  csbxp  5087  csbdm  5203  issubc  15082  nbgraopALT  24247  csbcom2fi  30462  bj-sbeq  33950  bj-sbceqgALT  33951  bj-sels  34002