Theorem sbcel2 3989

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 3983	. . 3 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3546	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = B )
3	2	eleq1d 2686	. . 3 |- ( A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) )
4	1, 3	syl5bb 272	. 2 |- ( A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
5		sbcex 3445	. . . 4 |- ( [. A / x ]. B e. C -> A e. _V )
6	5	con3i 150	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B e. C )
7		noel 3919	. . . 4 |- -. B e. (/)
8		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
9	8	eleq2d 2687	. . . 4 |- ( -. A e. _V -> ( B e. [_ A / x ]_ C <-> B e. (/) ) )
10	7, 9	mtbiri 317	. . 3 |- ( -. A e. _V -> -. B e. [_ A / x ]_ C )
11	6, 10	2falsed 366	. 2 |- ( -. A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
12	4, 11	pm2.61i 176	1 |- ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 196   e. wcel 1990   _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:  csbcom  3994  sbccsb  4004  sbnfc2  4007  csbab  4008  sbcssg  4085  csbuni  4466  csbxp  5200  csbdm  5318  issubc  16495  esum2dlem  30154  bj-sbeq  32896  bj-sbceqgALT  32897  bj-sels  32950  f1omptsnlem  33183  csbcom2fi  33934  disjinfi  39380  iccelpart  41369