Theorem sbcel2g 3145

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

Assertion

Ref	Expression
sbcel2g	|- ( A e. V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )

Distinct variable group:   x, B

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

Proof of Theorem sbcel2g

Step	Hyp	Ref	Expression
1		sbcel12g 3139	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
2		csbconstg 3138	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eleq1d 2298	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) )
4	1, 3	bitrd 188	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   [.wsbc 3028   [_csb 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbcomg  3147  sbccsbg  3153  sbnfc2  3185  csbabg  3186  sbcssg  3600  csbunig  3895  csbxpg  4799  csbdmg  4916  csbrng  5189  bj-sels  16235