Theorem sbcel2g 3118

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 3112	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
2		csbconstg 3111	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eleq1d 2275	. 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 2177   [.wsbc 3002   [_csb 3097

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-csb 3098

This theorem is referenced by:  csbcomg  3120  sbccsbg  3126  sbnfc2  3158  csbabg  3159  sbcssg  3573  csbunig  3864  csbxpg  4764  csbdmg  4881  csbrng  5153  bj-sels  15988