Theorem sbcel2g 2871

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 2865	. 2 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
2		csbconstg 2864	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eleq1d 2106	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) )
4	1, 3	bitrd 177	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   [.wsbc 2764   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  csbcomg  2873  sbccsbg  2878  sbnfc2  2906  csbabg  2907  sbcssg  3330  csbunig  3588  csbxpg  4421  csbdmg  4529  csbrng  4782  bj-sels  10034