Theorem sbcel2 3790

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 3784	. . 3 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3388	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3	2	eleq1d 2524	. . 3 |- ( A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) )
4	1, 3	syl5bb 265	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
5		sbcex 3289	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
6	5	con3i 142	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
7		noel 3747	. . . 4 |- -. B e. (/)
8		csbprc 3782	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
9	8	eleq2d 2525	. . . 4 |- ( -. A e. _V -> ( B e. [_ A / x ]_ C <-> B e. (/) ) )
10	7, 9	mtbiri 309	. . 3 |- ( -. A e. _V -> -. B e. [_ A / x ]_ C )
11	6, 10	2falsed 357	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
12	4, 11	pm2.61i 169	1 |- ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 189   e. wcel 1898   _Vcvv 3057   [.wsbc 3279   [_csb 3375   (/)c0 3743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744

This theorem is referenced by:  csbcom  3795  sbccsb  3805  sbnfc2  3808  csbab  3809  sbcssg  3892  csbuni  4240  csbxp  4935  csbdm  5048  issubc  15789  nbgraopALT  25201  esum2dlem  28962  bj-sbeq  31548  bj-sbceqgALT  31549  bj-sels  31601  f1omptsnlem  31783  csbcom2fi  32414  disjinfi  37506  iccelpart  38785