Theorem sbcel1g 3805

Description: Move proper substitution in and out of a membership relation. Note that the scope of [. A / x ]. is the wff B e. C, whereas the scope of [_ A / x ]_ is the class B. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
sbcel1g	|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Distinct variable group:   x, C

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

Proof of Theorem sbcel1g

Step	Hyp	Ref	Expression
1		sbcel12 3801	. 2 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
2		csbconstg 3409	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	eleq2d 2493	. 2 |- ( A e. V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> [_ A / x ]_ B e. C ) )
4	1, 3	syl5bb 261	1 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )

Syntax hints:   -> wi 4   <-> wb 188   e. wcel 1869   [.wsbc 3300   [_csb 3396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763

This theorem is referenced by:  rspcsbela  3824  fprodcllemf  14005  wunnat  15854  catcfuccl  15997  nbgraopALT  25144  rusgrasn  25665  esumpfinvalf  28899  esum2dlem  28915  measiuns  29041  bj-sbel1  31470  finixpnum  31850  renegclALT  32460  cdlemk35s  34429  ellimcabssub0  37523