Theorem sbcel2gOLD 38755

Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbcel2 3989 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcel2gOLD	|- ( 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 sbcel2gOLD

Step	Hyp	Ref	Expression
1		sbcel12gOLD 38754	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
2		csbconstg 3546	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eleq1d 2686	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) )
4	1, 3	bitrd 268	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   [.wsbc 3435   [_csb 3533

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-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

This theorem is referenced by:  sbcssOLD  38756  csbabgOLD  39050  csbunigOLD  39051  csbxpgOLD  39053  csbrngOLD  39056  sbcssgVD  39119  csbingVD  39120  csbunigVD  39134