Theorem sbcel2 3794

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 3786	. . 3 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3411	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3	2	eleq1d 2523	. . 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 3304	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
6	5	con3i 135	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
7		noel 3752	. . . 4 |- -. B e. (/)
8		csbprc 3784	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
9	8	eleq2d 2524	. . . 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 1758   _Vcvv 3078   [.wsbc 3294   [_csb 3398   (/)c0 3748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749

This theorem is referenced by:  csbcom  3800  sbccsb  3812  sbnfc2  3817  csbab  3818  sbcssg  3901  csbuni  4230  csbxp  5029  csbdm  5145  issubc  14871  csbcom2fi  29109  bj-sbeq  32760  bj-sbceqgALT  32761  bj-sels  32812